Section inv_eval_in_domain_proof.
Variable R Y : Ens.
Hypothesis i : IN Y (Range R).
Let X := inverse_EV R Y.
Let step0 : Range_prop R Y.
Proof Range_pr2 R Y i.
Let step1 : IN (PAIR X Y) R.
Proof inverse_EV_pr R Y step0.
Proof Domain_th1 R X Y step1.
End inv_eval_in_domain_proof.

Theorem inverse_EV_in_domain :
forall (R Y : Ens) (i : IN Y (Range R)), IN (inverse_EV R Y) (Domain R).