Theorem isorel 5858

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 5268	. . 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 4037	. . . 4 |- ( x = C -> ( x R y <-> C R y ) )
4		fveq2 5561	. . . . 5 |- ( x = C -> ( H ` x ) = ( H ` C ) )
5	4	breq1d 4044	. . . 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 4038	. . . 4 |- ( y = D -> ( C R y <-> C R D ) )
8		fveq2 5561	. . . . 5 |- ( y = D -> ( H ` y ) = ( H ` D ) )
9	8	breq2d 4046	. . . 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 2881	. 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 1364   e. wcel 2167   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-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-isom 5268

This theorem is referenced by:  isoresbr  5859  isoini  5868  isopolem  5872  isosolem  5874  smoiso  6369  isotilem  7081  supisolem  7083  ordiso2  7110  leisorel  10946  zfz1isolemiso  10948  seq3coll  10951