Theorem isoeq5 5945

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 5573	. . 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 5335	. 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 5335	. 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 1397   A.wral 2510   class class class wbr 4088   -1-1-onto->wf1o 5325   ` cfv 5326   Isom wiso 5327

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-isom 5335

This theorem is referenced by:  isores3  5955  ordiso  7234  zfz1isolem1  11103  zfz1iso  11104