-- Mathlib/Geometry/Manifold/BumpFunction.lean
theorem image_eq_inter_preimage_of_subset_support {s : Set M} (hs : s ⊆ support f) :
    extChartAt I c '' s =
      closedBall (extChartAt I c c) f.rOut ∩ range I ∩ (extChartAt I c).symm ⁻¹' s

Proof state:
case intro.h₁.refine_2
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ (extChartAt I c).target ∩ ↑(extChartAt I c).symm ⁻¹' s ⊆ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
exact inter_subset_right

Result:
no goals

Proof state:
case intro.h₁.refine_1
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ range ↑I

Tactic:
rintro _ ⟨x, -, rfl⟩

Result:
case intro.h₁.refine_1.intro.intro
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
x : M
⊢ ↑(extChartAt I c) x ∈ range ↑I

Proof state:
case intro.h₁.refine_2
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
rw [(extChartAt I c).image_eq_target_inter_inv_preimage hse]

Result:
case intro.h₁.refine_2
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ (extChartAt I c).target ∩ ↑(extChartAt I c).symm ⁻¹' s ⊆ ↑(extChartAt I c).symm ⁻¹' s

Proof state:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hs : s ⊆ support ↑f
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
rw [support_eq_inter_preimage, subset_inter_iff, ← extChartAt_source I, ← image_subset_iff] at hs

Result:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hs : s ⊆ (extChartAt I c).source ∧ ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Proof state:
case intro.h₁
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
refine subset_inter (subset_inter (hsf.trans ball_subset_closedBall) ?_) ?_

Result:
case intro.h₁.refine_1
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ range ↑I

case intro.h₁.refine_2
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ ↑(extChartAt I c).symm ⁻¹' s

Proof state:
case intro
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
apply Subset.antisymm

Result:
case intro.h₁
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

case intro.h₂
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s ⊆ ↑(extChartAt I c) '' s

Proof state:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hs : s ⊆ (extChartAt I c).source ∧ ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
obtain ⟨hse, hsf⟩ := hs

Result:
case intro
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s