Theorem isoeq5 5796

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 5443	. . 3 |- ( B = C -> ( H : A -1-1-onto-> B <-> H : A -1-1-onto-> C ) )
2	1	anbi1d 465	. 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 5217	. 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 5217	. 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 223	1 |- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   A.wral 2453   class class class wbr 3998   -1-1-onto->wf1o 5207   ` cfv 5208   Isom wiso 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-in 3133  df-ss 3140  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-isom 5217

This theorem is referenced by:  isores3  5806  ordiso  7025  zfz1isolem1  10786  zfz1iso  10787