image_inv_closed

forall (G : Sort u), forall (oneG : G), forall (mulG : forall (a : G), forall (b : G), G), forall (invG : forall (a : G), G), forall (H : Sort v), forall (oneH : H), forall (mulH : forall (a : H), forall (b : H), H), forall (invH : forall (a : H), H), forall (f : forall (x : G), H), forall (hom_args : @GroupHomLawArgs.{u,v} G oneG mulG invG H oneH mulH invH f), forall (y : H), forall (hy : @ImagePred.{u,v} G H f y), @ImagePred.{u,v} G H f (invH y)

fun G => fun oneG => fun mulG => fun invG => fun H => fun oneH => fun mulH => fun invH => fun f => fun hom_args => fun y => fun hy => hy (@ImagePred.{u,v} G H f (invH y)) (fun (a : G) => fun (hay : @Eq.{v} H (f a) y) => fun (P : Prop) => fun (mk : forall (c : G), forall (h : @Eq.{v} H (f c) (invH y)), P) => mk (invG a) (@eq_trans.{v} H (f (invG a)) (invH (f a)) (invH y) (@hom_inv.{u,v} G oneG mulG invG H oneH mulH invH f hom_args a) (@eq_congr_arg.{v,v} H H invH (f a) y hay)))