Theorem isoeq3 5704

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

Assertion

Ref	Expression
isoeq3	|- ( S = T -> ( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) )

Proof of Theorem isoeq3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 3931	. . . . 5 |- ( S = T -> ( ( H ` x ) S ( H ` y ) <-> ( H ` x ) T ( H ` y ) ) )
2	1	bibi2d 231	. . . 4 |- ( S = T -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x R y <-> ( H ` x ) T ( H ` y ) ) ) )
3	2	2ralbidv 2459	. . 3 |- ( S = T -> ( A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> A. x e. A A. y e. A ( x R y <-> ( H ` x ) T ( H ` y ) ) ) )
4	3	anbi2d 459	. 2 |- ( S = T -> ( ( 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-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) T ( H ` y ) ) ) ) )
5		df-isom 5132	. 2 |- ( 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 ) ) ) )
6		df-isom 5132	. 2 |- ( H Isom R , T ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) T ( H ` y ) ) ) )
7	4, 5, 6	3bitr4g 222	1 |- ( S = T -> ( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   A.wral 2416   class class class wbr 3929   -1-1-onto->wf1o 5122   ` cfv 5123   Isom wiso 5124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135  df-ral 2421  df-br 3930  df-isom 5132

This theorem is referenced by: (None)