Theorem brinxp2 3237

Description: Intersection of binary relation with cross product.

Assertion

Ref	Expression
brinxp2	|- (B e. S -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))

Proof of Theorem brinxp2

Step	Hyp	Ref	Expression
1		opelxpg 3222	. . 3 |- (B e. S -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
2	1	anbi2d 618	. 2 |- (B e. S -> ((ARB /\ <.A, B>. e. (C X. D)) <-> (ARB /\ (A e. C /\ B e. D))))
3		elin 2210	. . 3 |- (<.A, B>. e. (R i^i (C X. D)) <-> (<.A, B>. e. R /\ <.A, B>. e. (C X. D)))
4		df-br 2625	. . 3 |- (A(R i^i (C X. D))B <-> <.A, B>. e. (R i^i (C X. D)))
5		df-br 2625	. . . 4 |- (ARB <-> <.A, B>. e. R)
6	5	anbi1i 483	. . 3 |- ((ARB /\ <.A, B>. e. (C X. D)) <-> (<.A, B>. e. R /\ <.A, B>. e. (C X. D)))
7	3, 4, 6	3bitr4 183	. 2 |- (A(R i^i (C X. D))B <-> (ARB /\ <.A, B>. e. (C X. D)))
8		3anrot 782	. . 3 |- ((ARB /\ A e. C /\ B e. D) <-> (A e. C /\ B e. D /\ ARB))
9		3anass 781	. . 3 |- ((ARB /\ A e. C /\ B e. D) <-> (ARB /\ (A e. C /\ B e. D)))
10	8, 9	bitr3 175	. 2 |- ((A e. C /\ B e. D /\ ARB) <-> (ARB /\ (A e. C /\ B e. D)))
11	2, 7, 10	3bitr4g 557	1 |- (B e. S -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   e. wcel 960   i^i cin 2049  <.cop 2415   class class class wbr 2624   X. cxp 3174

This theorem is referenced by:  brinxp 3238  fncnv 3567  inposet 10477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190

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