Theorem isoeq5 5584

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

Assertion

Ref	Expression
isoeq5	|- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )

Proof of Theorem isoeq5

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

Step	Hyp	Ref	Expression
1		f1oeq3 5246	. . 3 |- ( B = C -> ( H : A -1-1-onto-> B <-> H : A -1-1-onto-> C ) )
2	1	anbi1d 453	. 2 |- ( B = C -> ( ( 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-> C /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) )
3		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 ) ) ) )
4		df-isom 5024	. 2 |- ( H Isom R , S ( A , C ) <-> ( H : A -1-1-onto-> C /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
5	2, 3, 4	3bitr4g 221	1 |- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )

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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-isom 5024

This theorem is referenced by:  isores3  5594  ordiso  6729  zfz1isolem1  10245  zfz1iso  10246