Canonical pcore_igFun pi := [igFun by pcore_sub pi & morphim_pcore pi].
Canonical pcore_gFun pi := [gFun by morphim_pcore pi].
Canonical pcore_pgFun pi := [pgFun by morphim_pcore pi].

(* Specifications: *)
Lemma pcore_char pi gT (G : {group gT}) : 'O_pi(G) \char G.
Lemma quotient_pcore_mod pi gT (G : {group gT}) (B : {set gT}) :
pcore_mod G pi B / B = 'O_pi(G / B).
Lemma pcore_mod_res pi gT rT (D : {group gT}) (f : {morphism D >-> rT}) :
f @* pcore_mod D pi (F _ D) \subset pcore_mod (f @* D) pi (F _ (f @* D)).