Theorem isorel 5468

Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Assertion

Ref	Expression
isorel	|- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D <-> ( H ` C ) S ( H ` D ) ) )

Proof of Theorem isorel

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

Step	Hyp	Ref	Expression
1		df-isom 4931	. . 3 |- ( 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 ) ) ) )
2	1	simprbi 269	. 2 |- ( H Isom R , S ( A , B ) -> A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) )
3		breq1 3788	. . . 4 |- ( x = C -> ( x R y <-> C R y ) )
4		fveq2 5198	. . . . 5 |- ( x = C -> ( H ` x ) = ( H ` C ) )
5	4	breq1d 3795	. . . 4 |- ( x = C -> ( ( H ` x ) S ( H ` y ) <-> ( H ` C ) S ( H ` y ) ) )
6	3, 5	bibi12d 233	. . 3 |- ( x = C -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( C R y <-> ( H ` C ) S ( H ` y ) ) ) )
7		breq2 3789	. . . 4 |- ( y = D -> ( C R y <-> C R D ) )
8		fveq2 5198	. . . . 5 |- ( y = D -> ( H ` y ) = ( H ` D ) )
9	8	breq2d 3797	. . . 4 |- ( y = D -> ( ( H ` C ) S ( H ` y ) <-> ( H ` C ) S ( H ` D ) ) )
10	7, 9	bibi12d 233	. . 3 |- ( y = D -> ( ( C R y <-> ( H ` C ) S ( H ` y ) ) <-> ( C R D <-> ( H ` C ) S ( H ` D ) ) ) )
11	6, 10	rspc2v 2713	. 2 |- ( ( C e. A /\ D e. A ) -> ( A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) -> ( C R D <-> ( H ` C ) S ( H ` D ) ) ) )
12	2, 11	mpan9 275	1 |- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D <-> ( H ` C ) S ( H ` D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   class class class wbr 3785   -1-1-onto->wf1o 4921   ` cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-isom 4931

This theorem is referenced by:  isoresbr  5469  isoini  5477  isopolem  5481  isosolem  5483  smoiso  5940  isotilem  6419  supisolem  6421  ordiso2  6446