Theorem brab2a 4657

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 108	. . . . 5 |- ( ( ( x e. C /\ y e. D ) /\ ph ) -> ( x e. C /\ y e. D ) )
2	1	ssopab2i 4255	. . . 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 4610	. . . 4 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
5	2, 3, 4	3sstr4i 3183	. . 3 |- R C_ ( C X. D )
6	5	brel 4656	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
7		df-br 3983	. . . 4 |- ( A R B <-> <. A , B >. e. R )
8	3	eleq2i 2233	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
9	7, 8	bitri 183	. . 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 4243	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
12	9, 11	syl5bb 191	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
13	6, 12	biadan2 452	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   {copab 4042   X. cxp 4602

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610

This theorem is referenced by:  lmbr  12853