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Theorem xpkexg 4288
 Description: The Kuratowski cross product of two sets is a set. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
xpkexg ((A V B W) → (A ×k B) V)

Proof of Theorem xpkexg
StepHypRef Expression
1 cnvkxpk 4276 . . 3 k(V ×k A) = (A ×k V)
2 xpkvexg 4285 . . . 4 (A V → (V ×k A) V)
3 cnvkexg 4286 . . . 4 ((V ×k A) V → k(V ×k A) V)
42, 3syl 15 . . 3 (A Vk(V ×k A) V)
51, 4syl5eqelr 2438 . 2 (A V → (A ×k V) V)
6 xpkvexg 4285 . 2 (B W → (V ×k B) V)
7 inxpk 4277 . . . 4 ((A ×k V) ∩ (V ×k B)) = ((A ∩ V) ×k (V ∩ B))
8 inv1 3577 . . . . 5 (A ∩ V) = A
9 incom 3448 . . . . . 6 (V ∩ B) = (B ∩ V)
10 inv1 3577 . . . . . 6 (B ∩ V) = B
119, 10eqtri 2373 . . . . 5 (V ∩ B) = B
128, 11xpkeq12i 4203 . . . 4 ((A ∩ V) ×k (V ∩ B)) = (A ×k B)
137, 12eqtri 2373 . . 3 ((A ×k V) ∩ (V ×k B)) = (A ×k B)
14 inexg 4100 . . 3 (((A ×k V) V (V ×k B) V) → ((A ×k V) ∩ (V ×k B)) V)
1513, 14syl5eqelr 2438 . 2 (((A ×k V) V (V ×k B) V) → (A ×k B) V)
165, 6, 15syl2an 463 1 ((A V B W) → (A ×k B) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208   ×k cxpk 4174  ◡kccnvk 4175 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-xp 4079  ax-cnv 4080  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-rex 2620  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  df-cnvk 4186 This theorem is referenced by:  xpkex  4289  uni1exg  4292  imakexg  4299  pw1exg  4302  pwexg  4328
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