Metamath Proof Explorer


Theorem elghomOLD

Description: Obsolete version of isghm as of 15-Mar-2020. Membership in the set of group homomorphisms from G to H . (Contributed by Paul Chapman, 3-Mar-2008) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses elghomOLD.1 X = ran G
elghomOLD.2 W = ran H
Assertion elghomOLD G GrpOp H GrpOp F G GrpOpHom H F : X W x X y X F x H F y = F x G y

Proof

Step Hyp Ref Expression
1 elghomOLD.1 X = ran G
2 elghomOLD.2 W = ran H
3 eqid f | f : ran G ran H x ran G y ran G f x H f y = f x G y = f | f : ran G ran H x ran G y ran G f x H f y = f x G y
4 3 elghomlem2OLD G GrpOp H GrpOp F G GrpOpHom H F : ran G ran H x ran G y ran G F x H F y = F x G y
5 1 2 feq23i F : X W F : ran G ran H
6 1 raleqi y X F x H F y = F x G y y ran G F x H F y = F x G y
7 1 6 raleqbii x X y X F x H F y = F x G y x ran G y ran G F x H F y = F x G y
8 5 7 anbi12i F : X W x X y X F x H F y = F x G y F : ran G ran H x ran G y ran G F x H F y = F x G y
9 4 8 bitr4di G GrpOp H GrpOp F G GrpOpHom H F : X W x X y X F x H F y = F x G y