Theorem isoeq1 5925

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 5560	. . 3 |- ( H = G -> ( H : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
2		fveq1 5626	. . . . . 6 |- ( H = G -> ( H ` x ) = ( G ` x ) )
3		fveq1 5626	. . . . . 6 |- ( H = G -> ( H ` y ) = ( G ` y ) )
4	2, 3	breq12d 4096	. . . . 5 |- ( H = G -> ( ( H ` x ) S ( H ` y ) <-> ( G ` x ) S ( G ` y ) ) )
5	4	bibi2d 232	. . . 4 |- ( H = G -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( x R y <-> ( G ` x ) S ( G ` y ) ) ) )
6	5	2ralbidv 2554	. . 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 473	. 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 5327	. 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 5327	. 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 223	1 |- ( H = G -> ( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   A.wral 2508   class class class wbr 4083   -1-1-onto->wf1o 5317   ` cfv 5318   Isom wiso 5319

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327

This theorem is referenced by:  isores1  5938  ordiso  7203  infrenegsupex  9789  zfz1isolem1  11062  zfz1iso  11063  infxrnegsupex  11774  relogiso  15547