Theorem isoeq2 5781

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 3991	. . . . 5 |- ( R = T -> ( x R y <-> x T y ) )
2	1	bibi1d 232	. . . 4 |- ( R = T -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x T y <-> ( H ` x ) S ( H ` y ) ) ) )
3	2	2ralbidv 2494	. . 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 461	. 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 5207	. 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 5207	. 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 222	1 |- ( R = T -> ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   A.wral 2448   class class class wbr 3989   -1-1-onto->wf1o 5197   ` cfv 5198   Isom wiso 5199

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-ral 2453  df-br 3990  df-isom 5207

This theorem is referenced by: (None)