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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 e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom H ) <-> ( F : X --> W /\ A. x e. X A. y e. 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 /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) } = { f | ( f : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( f ` x ) H ( f ` y ) ) = ( f ` ( x G y ) ) ) }
4 3 elghomlem2OLD 
 |-  ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom H ) <-> ( F : ran G --> ran H /\ A. x e. ran G A. y e. 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 
 |-  ( A. y e. X ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) <-> A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) )
7 1 6 raleqbii 
 |-  ( A. x e. X A. y e. X ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) <-> A. x e. ran G A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) )
8 5 7 anbi12i 
 |-  ( ( F : X --> W /\ A. x e. X A. y e. X ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) <-> ( F : ran G --> ran H /\ A. x e. ran G A. y e. ran G ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) )
9 4 8 bitr4di 
 |-  ( ( G e. GrpOp /\ H e. GrpOp ) -> ( F e. ( G GrpOpHom H ) <-> ( F : X --> W /\ A. x e. X A. y e. X ( ( F ` x ) H ( F ` y ) ) = ( F ` ( x G y ) ) ) ) )