Theorem inxpk 4278

Description: The intersection of two Kuratowski cross products. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
inxpk	|- ((A X.k B) i^i (C X.k D)) = ((A i^i C) X.k (B i^i D))

Proof of Theorem inxpk

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

Step	Hyp	Ref	Expression
1		inss1 3476	. . 3 |- ((A X.k B) i^i (C X.k D)) C_ (A X.k B)
2		xpkssvvk 4211	. . 3 |- (A X.k B) C_ (_V X.k _V)
3	1, 2	sstri 3282	. 2 |- ((A X.k B) i^i (C X.k D)) C_ (_V X.k _V)
4		xpkssvvk 4211	. 2 |- ((A i^i C) X.k (B i^i D)) C_ (_V X.k _V)
5		an4 797	. . 3 |- (((x e. A /\ y e. B) /\ (x e. C /\ y e. D)) <-> ((x e. A /\ x e. C) /\ (y e. B /\ y e. D)))
6		elin 3220	. . . 4 |- (<<x, y>> e. ((A X.k B) i^i (C X.k D)) <-> (<<x, y>> e. (A X.k B) /\ <<x, y>> e. (C X.k D)))
7		vex 2863	. . . . . 6 |- x e. _V
8		vex 2863	. . . . . 6 |- y e. _V
9	7, 8	opkelxpk 4249	. . . . 5 |- (<<x, y>> e. (A X.k B) <-> (x e. A /\ y e. B))
10	7, 8	opkelxpk 4249	. . . . 5 |- (<<x, y>> e. (C X.k D) <-> (x e. C /\ y e. D))
11	9, 10	anbi12i 678	. . . 4 |- ((<<x, y>> e. (A X.k B) /\ <<x, y>> e. (C X.k D)) <-> ((x e. A /\ y e. B) /\ (x e. C /\ y e. D)))
12	6, 11	bitri 240	. . 3 |- (<<x, y>> e. ((A X.k B) i^i (C X.k D)) <-> ((x e. A /\ y e. B) /\ (x e. C /\ y e. D)))
13	7, 8	opkelxpk 4249	. . . 4 |- (<<x, y>> e. ((A i^i C) X.k (B i^i D)) <-> (x e. (A i^i C) /\ y e. (B i^i D)))
14		elin 3220	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
15		elin 3220	. . . . 5 |- (y e. (B i^i D) <-> (y e. B /\ y e. D))
16	14, 15	anbi12i 678	. . . 4 |- ((x e. (A i^i C) /\ y e. (B i^i D)) <-> ((x e. A /\ x e. C) /\ (y e. B /\ y e. D)))
17	13, 16	bitri 240	. . 3 |- (<<x, y>> e. ((A i^i C) X.k (B i^i D)) <-> ((x e. A /\ x e. C) /\ (y e. B /\ y e. D)))
18	5, 12, 17	3bitr4i 268	. 2 |- (<<x, y>> e. ((A X.k B) i^i (C X.k D)) <-> <<x, y>> e. ((A i^i C) X.k (B i^i D)))
19	3, 4, 18	eqrelkriiv 4214	1 |- ((A X.k B) i^i (C X.k D)) = ((A i^i C) X.k (B i^i D))

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860   i^i cin 3209  <<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-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

This theorem is referenced by:  xpkexg  4289