Theorem isoeq1 5746

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

Assertion

Ref	Expression
isoeq1	|- ( H = G -> ( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )

Proof of Theorem isoeq1

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

Step	Hyp	Ref	Expression
1		f1oeq1 5400	. . 3 |- ( H = G -> ( H : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
2		fveq1 5464	. . . . . 6 |- ( H = G -> ( H ` x ) = ( G ` x ) )
3		fveq1 5464	. . . . . 6 |- ( H = G -> ( H ` y ) = ( G ` y ) )
4	2, 3	breq12d 3978	. . . . 5 |- ( H = G -> ( ( H ` x ) S ( H ` y ) <-> ( G ` x ) S ( G ` y ) ) )
5	4	bibi2d 231	. . . 4 |- ( H = G -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x R y <-> ( G ` x ) S ( G ` y ) ) ) )
6	5	2ralbidv 2481	. . 3 |- ( H = G -> ( 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 R y <-> ( G ` x ) S ( G ` y ) ) ) )
7	1, 6	anbi12d 465	. 2 |- ( H = G -> ( ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) <-> ( G : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( G ` x ) S ( G ` y ) ) ) ) )
8		df-isom 5176	. 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 ) ) ) )
9		df-isom 5176	. 2 |- ( G Isom R , S ( A , B ) <-> ( G : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( G ` x ) S ( G ` y ) ) ) )
10	7, 8, 9	3bitr4g 222	1 |- ( H = G -> ( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   A.wral 2435   class class class wbr 3965   -1-1-onto->wf1o 5166   ` cfv 5167   Isom wiso 5168

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176

This theorem is referenced by:  isores1  5759  ordiso  6970  infrenegsupex  9488  zfz1isolem1  10693  zfz1iso  10694  infxrnegsupex  11142  relogiso  13154