Theorem opelxpg 3222

Description: Ordered pair membership in a cross product.

Assertion

Ref	Expression
opelxpg	|- (B e. R -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))

Proof of Theorem opelxpg

Step	Hyp	Ref	Expression
1		opeq2 2492	. . 3 |- (x = B -> <.A, x>. = <.A, B>.)
2	1	eleq1d 1543	. 2 |- (x = B -> (<.A, x>. e. (C X. D) <-> <.A, B>. e. (C X. D)))
3		eleq1 1537	. . 3 |- (x = B -> (x e. D <-> B e. D))
4	3	anbi2d 618	. 2 |- (x = B -> ((A e. C /\ x e. D) <-> (A e. C /\ B e. D)))
5		visset 1816	. . 3 |- x e. V
6	5	opelxp 3220	. 2 |- (<.A, x>. e. (C X. D) <-> (A e. C /\ x e. D))
7	2, 4, 6	vtoclbg 1851	1 |- (B e. R -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  <.cop 2415   X. cxp 3174

This theorem is referenced by:  opelxpi 3223  brelg 3228  brinxp2 3237  ndmoprg 4049  ltxrt 5507

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-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-opab 2672  df-xp 3190

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