Theorem brab2a 4440

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)

Hypotheses

Ref	Expression
brab2a.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
brab2a.2	|- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) }

Assertion

Ref	Expression
brab2a	|- ( 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 brab2a

Step	Hyp	Ref	Expression
1		simpl 107	. . . . 5 |- ( ( ( x e. C /\ y e. D ) /\ ph ) -> ( x e. C /\ y e. D ) )
2	1	ssopab2i 4061	. . . 4 |- { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } C_ { <. x , y >. | ( x e. C /\ y e. D ) }
3		brab2a.2	. . . 4 |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) }
4		df-xp 4398	. . . 4 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
5	2, 3, 4	3sstr4i 3048	. . 3 |- R C_ ( C X. D )
6	5	brel 4439	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
7		df-br 3807	. . . 4 |- ( A R B <-> <. A , B >. e. R )
8	3	eleq2i 2149	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
9	7, 8	bitri 182	. . 3 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
10		brab2a.1	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
11	10	opelopab2a 4049	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
12	9, 11	syl5bb 190	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
13	6, 12	biadan2 444	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   <.cop 3420   class class class wbr 3806   {copab 3859   X. cxp 4390

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398

This theorem is referenced by: (None)