Theorem opelxpg 2454

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 1877	. . 3 |- (x = B -> <.A, x>. = <.A, B>.)
2	1	eleq1d 1155	. 2 |- (x = B -> (<.A, x>. e. (C X. D) <-> <.A, B>. e. (C X. D)))
3		eleq1 1149	. . 3 |- (x = B -> (x e. D <-> B e. D))
4	3	anbi2d 468	. 2 |- (x = B -> ((A e. C /\ x e. D) <-> (A e. C /\ B e. D)))
5		visset 1350	. . 3 |- x e. V
6	5	opelxp 2452	. 2 |- (<.A, x>. e. (C X. D) <-> (A e. C /\ x e. D))
7	2, 4, 6	vtoclbg 1384	1 |- (B e. R -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810   X. cxp 2408

This theorem is referenced by:  opelxpi 2455  brelg 2458  ndmoprg 3057

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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