Theorem brinxp2ALTV 34034

Description: Intersection with cross product binary relation . (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brinxp2ALTV	|- ( C ( R i^i ( A X. B ) ) D <-> ( ( C e. A /\ D e. B ) /\ C R D ) )

Proof of Theorem brinxp2ALTV

Step	Hyp	Ref	Expression
1		brin 4704	. 2 |- ( C ( R i^i ( A X. B ) ) D <-> ( C R D /\ C ( A X. B ) D ) )
2		ancom 466	. 2 |- ( ( C R D /\ C ( A X. B ) D ) <-> ( C ( A X. B ) D /\ C R D ) )
3		brxp 5147	. . 3 |- ( C ( A X. B ) D <-> ( C e. A /\ D e. B ) )
4	3	anbi1i 731	. 2 |- ( ( C ( A X. B ) D /\ C R D ) <-> ( ( C e. A /\ D e. B ) /\ C R D ) )
5	1, 2, 4	3bitri 286	1 |- ( C ( R i^i ( A X. B ) ) D <-> ( ( C e. A /\ D e. B ) /\ C R D ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   i^i cin 3573   class class class wbr 4653   X. cxp 5112

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-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

This theorem is referenced by:  brres2  34035  inxpss  34082  inxpss3  34084  idinxpssinxp2  34089  opelinxp  34111  inxp2  34129