Theorem brab2ga 4433

Description: The law of concretion for a binary relation. See brab2a 4411 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 4432	. . . 4 |- { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } C_ ( C X. D )
3	1, 2	eqsstri 3029	. . 3 |- R C_ ( C X. D )
4	3	brel 4410	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
5		df-br 3786	. . . 4 |- ( A R B <-> <. A , B >. e. R )
6	1	eleq2i 2145	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
7	5, 6	bitri 182	. . 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 4020	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
10	7, 9	syl5bb 190	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
11	4, 10	biadan2 443	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   class class class wbr 3785   {copab 3838   X. cxp 4361

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-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  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-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369

This theorem is referenced by:  reapval  7676  ltxr  8849