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Metamath Proof Explorer

Theorem isoeq5

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004)

Ref Expression
Assertion isoeq5 
|- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )

Proof

Step Hyp Ref Expression
1 f1oeq3 
 |-  ( B = C -> ( H : A -1-1-onto-> B <-> H : A -1-1-onto-> C ) )
2 1 anbi1d 
 |-  ( B = C -> ( ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) <-> ( H : A -1-1-onto-> C /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) )
3 df-isom 
 |-  ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
4 df-isom 
 |-  ( H Isom R , S ( A , C ) <-> ( H : A -1-1-onto-> C /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
5 2 3 4 3bitr4g 
 |-  ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )