Theorem isoeq1 5851

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 5495	. . 3 |- ( H = G -> ( H : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
2		fveq1 5560	. . . . . 6 |- ( H = G -> ( H ` x ) = ( G ` x ) )
3		fveq1 5560	. . . . . 6 |- ( H = G -> ( H ` y ) = ( G ` y ) )
4	2, 3	breq12d 4047	. . . . 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 2521	. . 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 5268	. 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 5268	. 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 1364   A.wral 2475   class class class wbr 4034   -1-1-onto->wf1o 5258   ` cfv 5259   Isom wiso 5260

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268

This theorem is referenced by:  isores1  5864  ordiso  7111  infrenegsupex  9685  zfz1isolem1  10949  zfz1iso  10950  infxrnegsupex  11445  relogiso  15193