Theorem isorel 5944

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 5333	. . 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 275	. 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 4089	. . . 4 |- ( x = C -> ( x R y <-> C R y ) )
4		fveq2 5635	. . . . 5 |- ( x = C -> ( H ` x ) = ( H ` C ) )
5	4	breq1d 4096	. . . 4 |- ( x = C -> ( ( H ` x ) S ( H ` y ) <-> ( H ` C ) S ( H ` y ) ) )
6	3, 5	bibi12d 235	. . 3 |- ( x = C -> ( ( x R y <-> ( H ` x ) S ( H ` y ) ) <-> ( C R y <-> ( H ` C ) S ( H ` y ) ) ) )
7		breq2 4090	. . . 4 |- ( y = D -> ( C R y <-> C R D ) )
8		fveq2 5635	. . . . 5 |- ( y = D -> ( H ` y ) = ( H ` D ) )
9	8	breq2d 4098	. . . 4 |- ( y = D -> ( ( H ` C ) S ( H ` y ) <-> ( H ` C ) S ( H ` D ) ) )
10	7, 9	bibi12d 235	. . 3 |- ( y = D -> ( ( C R y <-> ( H ` C ) S ( H ` y ) ) <-> ( C R D <-> ( H ` C ) S ( H ` D ) ) ) )
11	6, 10	rspc2v 2921	. 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 281	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 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4086   -1-1-onto->wf1o 5323   ` cfv 5324   Isom wiso 5325

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-isom 5333

This theorem is referenced by:  isoresbr  5945  isoini  5954  isopolem  5958  isosolem  5960  smoiso  6463  isotilem  7196  supisolem  7198  ordiso2  7225  leisorel  11091  zfz1isolemiso  11093  seq3coll  11096