Theorem cnvkxpk 4277

Description: The converse of a Kuratowski cross product. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
cnvkxpk	|- `'k(A X.k B) = (B X.k A)

Proof of Theorem cnvkxpk

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

Step	Hyp	Ref	Expression
1		cnvkssvvk 4276	. 2 |- `'k(A X.k B) C_ (_V X.k _V)
2		xpkssvvk 4211	. 2 |- (B X.k A) C_ (_V X.k _V)
3		ancom 437	. . 3 |- ((y e. A /\ x e. B) <-> (x e. B /\ y e. A))
4		vex 2863	. . . . 5 |- x e. _V
5		vex 2863	. . . . 5 |- y e. _V
6	4, 5	opkelcnvk 4251	. . . 4 |- (<<x, y>> e. `'k(A X.k B) <-> <<y, x>> e. (A X.k B))
7	5, 4	opkelxpk 4249	. . . 4 |- (<<y, x>> e. (A X.k B) <-> (y e. A /\ x e. B))
8	6, 7	bitri 240	. . 3 |- (<<x, y>> e. `'k(A X.k B) <-> (y e. A /\ x e. B))
9	4, 5	opkelxpk 4249	. . 3 |- (<<x, y>> e. (B X.k A) <-> (x e. B /\ y e. A))
10	3, 8, 9	3bitr4i 268	. 2 |- (<<x, y>> e. `'k(A X.k B) <-> <<x, y>> e. (B X.k A))
11	1, 2, 10	eqrelkriiv 4214	1 |- `'k(A X.k B) = (B X.k A)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  <<copk 4058   X._k cxpk 4175  `'_kccnvk 4176

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-ral 2620  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  df-cnvk 4187

This theorem is referenced by:  xpkexg  4289