Theorem isoeq3 6569

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 4655	. . . . 5 |- ( S = T -> ( ( H ` x ) S ( H ` y ) <-> ( H ` x ) T ( H ` y ) ) )
2	1	bibi2d 332	. . . 4 |- ( S = T -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x R y <-> ( H ` x ) T ( H ` y ) ) ) )
3	2	2ralbidv 2989	. . 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 740	. 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 5897	. 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 5897	. 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 303	1 |- ( S = T -> ( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   class class class wbr 4653   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ral 2917  df-br 4654  df-isom 5897

This theorem is referenced by:  fnwelem  7292  hartogslem1  8447  leiso  13243  gtiso  29478