Theorem inxp2 34129

Description: Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)

Assertion

Ref	Expression
inxp2	|- ( R i^i ( A X. B ) ) = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ x R y ) }

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

Proof of Theorem inxp2

Step	Hyp	Ref	Expression
1		relinxp 34069	. . 3 |- Rel ( R i^i ( A X. B ) )
2		dfrel4v 5584	. . 3 |- ( Rel ( R i^i ( A X. B ) ) <-> ( R i^i ( A X. B ) ) = { <. x , y >. | x ( R i^i ( A X. B ) ) y } )
3	1, 2	mpbi 220	. 2 |- ( R i^i ( A X. B ) ) = { <. x , y >. | x ( R i^i ( A X. B ) ) y }
4		brinxp2ALTV 34034	. . 3 |- ( x ( R i^i ( A X. B ) ) y <-> ( ( x e. A /\ y e. B ) /\ x R y ) )
5	4	opabbii 4717	. 2 |- { <. x , y >. | x ( R i^i ( A X. B ) ) y } = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ x R y ) }
6	3, 5	eqtri 2644	1 |- ( R i^i ( A X. B ) ) = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ x R y ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   {copab 4712   X. cxp 5112   Rel wrel 5119

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-ral 2917  df-rex 2918  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  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)