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Theorem inxpk 4277
 Description: The intersection of two Kuratowski cross products. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
inxpk ((A ×k B) ∩ (C ×k D)) = ((AC) ×k (BD))

Proof of Theorem inxpk
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3475 . . 3 ((A ×k B) ∩ (C ×k D)) (A ×k B)
2 xpkssvvk 4210 . . 3 (A ×k B) (V ×k V)
31, 2sstri 3281 . 2 ((A ×k B) ∩ (C ×k D)) (V ×k V)
4 xpkssvvk 4210 . 2 ((AC) ×k (BD)) (V ×k V)
5 an4 797 . . 3 (((x A y B) (x C y D)) ↔ ((x A x C) (y B y D)))
6 elin 3219 . . . 4 (⟪x, y ((A ×k B) ∩ (C ×k D)) ↔ (⟪x, y (A ×k B) x, y (C ×k D)))
7 vex 2862 . . . . . 6 x V
8 vex 2862 . . . . . 6 y V
97, 8opkelxpk 4248 . . . . 5 (⟪x, y (A ×k B) ↔ (x A y B))
107, 8opkelxpk 4248 . . . . 5 (⟪x, y (C ×k D) ↔ (x C y D))
119, 10anbi12i 678 . . . 4 ((⟪x, y (A ×k B) x, y (C ×k D)) ↔ ((x A y B) (x C y D)))
126, 11bitri 240 . . 3 (⟪x, y ((A ×k B) ∩ (C ×k D)) ↔ ((x A y B) (x C y D)))
137, 8opkelxpk 4248 . . . 4 (⟪x, y ((AC) ×k (BD)) ↔ (x (AC) y (BD)))
14 elin 3219 . . . . 5 (x (AC) ↔ (x A x C))
15 elin 3219 . . . . 5 (y (BD) ↔ (y B y D))
1614, 15anbi12i 678 . . . 4 ((x (AC) y (BD)) ↔ ((x A x C) (y B y D)))
1713, 16bitri 240 . . 3 (⟪x, y ((AC) ×k (BD)) ↔ ((x A x C) (y B y D)))
185, 12, 173bitr4i 268 . 2 (⟪x, y ((A ×k B) ∩ (C ×k D)) ↔ ⟪x, y ((AC) ×k (BD)))
193, 4, 18eqrelkriiv 4213 1 ((A ×k B) ∩ (C ×k D)) = ((AC) ×k (BD))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  ⟪copk 4057   ×k cxpk 4174 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 4078  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-xpk 4185 This theorem is referenced by:  xpkexg  4288
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