Theorem brab2a 4803

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 109	. . . . 5 |- ( ( ( x e. C /\ y e. D ) /\ ph ) -> ( x e. C /\ y e. D ) )
2	1	ssopab2i 4396	. . . 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 4755	. . . 4 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
5	2, 3, 4	3sstr4i 3279	. . 3 |- R C_ ( C X. D )
6	5	brel 4802	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
7		df-br 4110	. . . 4 |- ( A R B <-> <. A , B >. e. R )
8	3	eleq2i 2299	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
9	7, 8	bitri 184	. . 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 4383	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
12	9, 11	bitrid 192	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
13	6, 12	biadan2 456	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   <.cop 3692   class class class wbr 4109   {copab 4170   X. cxp 4747

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755

This theorem is referenced by:  lmbr  15078