Theorem brab2a 4728

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 4324	. . . 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 4681	. . . 4 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
5	2, 3, 4	3sstr4i 3234	. . 3 |- R C_ ( C X. D )
6	5	brel 4727	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
7		df-br 4045	. . . 4 |- ( A R B <-> <. A , B >. e. R )
8	3	eleq2i 2272	. . . 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 4311	. . 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 1373   e. wcel 2176   <.cop 3636   class class class wbr 4044   {copab 4104   X. cxp 4673

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681

This theorem is referenced by:  lmbr  14685