Theorem opkelxpkg 4248

Description: Kuratowski ordered pair membership in a Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkelxpkg	|- ((A e. V /\ B e. W) -> (<<A, B>> e. (C X.k D) <-> (A e. C /\ B e. D)))

Proof of Theorem opkelxpkg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xpk 4186	. 2 |- (C X.k D) = {z | E.xE.y(z = <<x, y>> /\ (x e. C /\ y e. D))}
2		eleq1 2413	. . 3 |- (x = A -> (x e. C <-> A e. C))
3	2	anbi1d 685	. 2 |- (x = A -> ((x e. C /\ y e. D) <-> (A e. C /\ y e. D)))
4		eleq1 2413	. . 3 |- (y = B -> (y e. D <-> B e. D))
5	4	anbi2d 684	. 2 |- (y = B -> ((A e. C /\ y e. D) <-> (A e. C /\ B e. D)))
6	1, 3, 5	opkelopkabg 4246	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. (C X.k D) <-> (A e. C /\ B e. D)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  <<copk 4058   X._k cxpk 4175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-xpk 4186

This theorem is referenced by:  opkelxpk  4249  sikss1c1c  4268  opkelimagekg  4272