Theorem brabgaf 29420

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)

Hypotheses

Ref	Expression
brabgaf.0	|- F/ x ps
brabgaf.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
brabgaf.2	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
brabgaf	|- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

Distinct variable groups:   x, y, A   x, B, y   ps, y

Allowed substitution hints:   ph( x, y)   ps( x)   R( x, y)   V( x, y)   W( x, y)

Proof of Theorem brabgaf

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( A R B <-> <. A , B >. e. R )
2		brabgaf.2	. . . 4 |- R = { <. x , y >. | ph }
3	2	eleq2i 2693	. . 3 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ph } )
4	1, 3	bitri 264	. 2 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ph } )
5		elopab 4983	. . 3 |- ( <. A , B >. e. { <. x , y >. | ph } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) )
6		elisset 3215	. . . 4 |- ( A e. V -> E. x x = A )
7		elisset 3215	. . . 4 |- ( B e. W -> E. y y = B )
8		eeanv 2182	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
9		nfe1 2027	. . . . . . 7 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
10		brabgaf.0	. . . . . . 7 |- F/ x ps
11	9, 10	nfbi 1833	. . . . . 6 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
12		nfe1 2027	. . . . . . . . 9 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
13	12	nfex 2154	. . . . . . . 8 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
14		nfv 1843	. . . . . . . 8 |- F/ y ps
15	13, 14	nfbi 1833	. . . . . . 7 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
16		opeq12 4404	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
17		copsexg 4956	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
18	17	eqcoms 2630	. . . . . . . . 9 |- ( <. x , y >. = <. A , B >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
19	16, 18	syl 17	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
20		brabgaf.1	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
21	19, 20	bitr3d 270	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
22	15, 21	exlimi 2086	. . . . . 6 |- ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
23	11, 22	exlimi 2086	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
24	8, 23	sylbir 225	. . . 4 |- ( ( E. x x = A /\ E. y y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
25	6, 7, 24	syl2an 494	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
26	5, 25	syl5bb 272	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
27	4, 26	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  fmptcof2  29457