Theorem xpkeq2 4200

Description: Equality theorem for Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
xpkeq2	|- (A = B -> (C X.k A) = (C X.k B))

Proof of Theorem xpkeq2

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

Step	Hyp	Ref	Expression
1		rexeq 2809	. . . 4 |- (A = B -> (E.z e. A x = <<y, z>> <-> E.z e. B x = <<y, z>>))
2	1	rexbidv 2636	. . 3 |- (A = B -> (E.y e. C E.z e. A x = <<y, z>> <-> E.y e. C E.z e. B x = <<y, z>>))
3		elxpk2 4198	. . 3 |- (x e. (C X.k A) <-> E.y e. C E.z e. A x = <<y, z>>)
4		elxpk2 4198	. . 3 |- (x e. (C X.k B) <-> E.y e. C E.z e. B x = <<y, z>>)
5	2, 3, 4	3bitr4g 279	. 2 |- (A = B -> (x e. (C X.k A) <-> x e. (C X.k B)))
6	5	eqrdv 2351	1 |- (A = B -> (C X.k A) = (C X.k B))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  E.wrex 2616  <<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-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-rex 2621  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:  xpkeq12  4201  xpkeq2i  4203  xpkeq2d  4206  xpkvexg  4286