Theorem isoeq5 5706

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 5358	. . 3 |- ( B = C -> ( H : A -1-1-onto-> B <-> H : A -1-1-onto-> C ) )
2	1	anbi1d 460	. 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 5132	. 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 5132	. 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 222	1 |- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   A.wral 2416   class class class wbr 3929   -1-1-onto->wf1o 5122   ` cfv 5123   Isom wiso 5124

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-isom 5132

This theorem is referenced by:  isores3  5716  ordiso  6921  zfz1isolem1  10583  zfz1iso  10584