Theorem brab2ga 4622

Description: The law of concretion for a binary relation. See brab2a 4600 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
brab2ga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
brab2ga.2	|- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) }

Assertion

Ref	Expression
brab2ga	|- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   ps, x, y

Allowed substitution hints:   ph( x, y)   R( x, y)

Proof of Theorem brab2ga

Step	Hyp	Ref	Expression
1		brab2ga.2	. . . 4 |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) }
2		opabssxp 4621	. . . 4 |- { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } C_ ( C X. D )
3	1, 2	eqsstri 3134	. . 3 |- R C_ ( C X. D )
4	3	brel 4599	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
5		df-br 3938	. . . 4 |- ( A R B <-> <. A , B >. e. R )
6	1	eleq2i 2207	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
7	5, 6	bitri 183	. . 3 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
8		brab2ga.1	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
9	8	opelopab2a 4195	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
10	7, 9	syl5bb 191	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
11	4, 10	biadan2 452	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   <.cop 3535   class class class wbr 3937   {copab 3996   X. cxp 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553

This theorem is referenced by:  reapval  8362  ltxr  9592