Theorem brinxp2 3231

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 3216	. . 3 |- (B e. S -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
2	1	anbi2d 616	. 2 |- (B e. S -> ((ARB /\ <.A, B>. e. (C X. D)) <-> (ARB /\ (A e. C /\ B e. D))))
3		elin 2207	. . 3 |- (<.A, B>. e. (R i^i (C X. D)) <-> (<.A, B>. e. R /\ <.A, B>. e. (C X. D)))
4		df-br 2620	. . 3 |- (A(R i^i (C X. D))B <-> <.A, B>. e. (R i^i (C X. D)))
5		df-br 2620	. . . 4 |- (ARB <-> <.A, B>. e. R)
6	5	anbi1i 481	. . 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 780	. . 3 |- ((ARB /\ A e. C /\ B e. D) <-> (A e. C /\ B e. D /\ ARB))
9		3anass 779	. . 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 555	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 775   e. wcel 958   i^i cin 2046  <.cop 2411   class class class wbr 2619   X. cxp 3168

This theorem is referenced by:  brinxp 3232  fncnv 3561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184

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