Theorem isorel 5811

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 5227	. . 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 4008	. . . 4 |- ( x = C -> ( x R y <-> C R y ) )
4		fveq2 5517	. . . . 5 |- ( x = C -> ( H ` x ) = ( H ` C ) )
5	4	breq1d 4015	. . . 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 4009	. . . 4 |- ( y = D -> ( C R y <-> C R D ) )
8		fveq2 5517	. . . . 5 |- ( y = D -> ( H ` y ) = ( H ` D ) )
9	8	breq2d 4017	. . . 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 2856	. 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 1353   e. wcel 2148   A.wral 2455   class class class wbr 4005   -1-1-onto->wf1o 5217   ` cfv 5218   Isom wiso 5219

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-isom 5227

This theorem is referenced by:  isoresbr  5812  isoini  5821  isopolem  5825  isosolem  5827  smoiso  6305  isotilem  7007  supisolem  7009  ordiso2  7036  leisorel  10819  zfz1isolemiso  10821  seq3coll  10824