Theorem brab2ga 4845

Description: The law of concretion for a binary relation. See brab2a 4823 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 4844	. . . 4 |- { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } C_ ( C X. D )
3	1, 2	eqsstri 3280	. . 3 |- R C_ ( C X. D )
4	3	brel 4822	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
5		df-br 4126	. . . 4 |- ( A R B <-> <. A , B >. e. R )
6	1	eleq2i 2305	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
7	5, 6	bitri 184	. . 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 4402	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
10	7, 9	bitrid 192	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
11	4, 10	biadan2 460	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3708   class class class wbr 4125   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775

This theorem is referenced by:  reapval  8894  ltxr  10156