Theorem isoeq2 6211

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

Assertion

Ref	Expression
isoeq2	|- ( R = T -> ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) )

Proof of Theorem isoeq2

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

Step	Hyp	Ref	Expression
1		breq 4404	. . . . 5 |- ( R = T -> ( x R y <-> x T y ) )
2	1	bibi1d 321	. . . 4 |- ( R = T -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x T y <-> ( H ` x ) S ( H ` y ) ) ) )
3	2	2ralbidv 2832	. . 3 |- ( R = 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 T y <-> ( H ` x ) S ( H ` y ) ) ) )
4	3	anbi2d 710	. 2 |- ( R = 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 T y <-> ( H ` x ) S ( H ` y ) ) ) ) )
5		df-isom 5591	. 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 5591	. 2 |- ( H Isom T , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x T y <-> ( H ` x ) S ( H ` y ) ) ) )
7	4, 5, 6	3bitr4g 292	1 |- ( R = T -> ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   A.wral 2737   class class class wbr 4402   -1-1-onto->wf1o 5581   ` cfv 5582   Isom wiso 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-cleq 2444  df-clel 2447  df-ral 2742  df-br 4403  df-isom 5591

This theorem is referenced by:  leiso  12622  gtiso  28281