Theorem isoeq2 5581

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 3847	. . . . 5 |- ( R = T -> ( x R y <-> x T y ) )
2	1	bibi1d 231	. . . 4 |- ( R = T -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x T y <-> ( H ` x ) S ( H ` y ) ) ) )
3	2	2ralbidv 2402	. . 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 452	. 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 5024	. 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 5024	. 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 221	1 |- ( R = T -> ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   A.wral 2359   class class class wbr 3845   -1-1-onto->wf1o 5014   ` cfv 5015   Isom wiso 5016

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-ral 2364  df-br 3846  df-isom 5024

This theorem is referenced by: (None)