Theorem isoeq1 5769

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 5421	. . 3 |- ( H = G -> ( H : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
2		fveq1 5485	. . . . . 6 |- ( H = G -> ( H ` x ) = ( G ` x ) )
3		fveq1 5485	. . . . . 6 |- ( H = G -> ( H ` y ) = ( G ` y ) )
4	2, 3	breq12d 3995	. . . . 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 2490	. . 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 5197	. 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 5197	. 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 1343   A.wral 2444   class class class wbr 3982   -1-1-onto->wf1o 5187   ` cfv 5188   Isom wiso 5189

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197

This theorem is referenced by:  isores1  5782  ordiso  7001  infrenegsupex  9532  zfz1isolem1  10753  zfz1iso  10754  infxrnegsupex  11204  relogiso  13444