Theorem brab2ga 4686

Description: The law of concretion for a binary relation. See brab2a 4664 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 4685	. . . 4 |- { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } C_ ( C X. D )
3	1, 2	eqsstri 3179	. . 3 |- R C_ ( C X. D )
4	3	brel 4663	. 2 |- ( A R B -> ( A e. C /\ B e. D ) )
5		df-br 3990	. . . 4 |- ( A R B <-> <. A , B >. e. R )
6	1	eleq2i 2237	. . . 4 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
7	5, 6	bitri 183	. . 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 4250	. . 3 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
10	7, 9	syl5bb 191	. 2 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ps ) )
11	4, 10	biadan2 453	1 |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989   {copab 4049   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617

This theorem is referenced by:  reapval  8495  ltxr  9732