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Theorem ixpint 8916
Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ixpint (𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem ixpint
StepHypRef Expression
1 ixpeq2 8902 . . 3 (∀𝑥𝐴 𝐵 = 𝑦𝐵 𝑦X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦)
2 intiin 5053 . . . 4 𝐵 = 𝑦𝐵 𝑦
32a1i 11 . . 3 (𝑥𝐴 𝐵 = 𝑦𝐵 𝑦)
41, 3mprg 3059 . 2 X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦
5 ixpiin 8915 . 2 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝑦 = 𝑦𝐵 X𝑥𝐴 𝑦)
64, 5eqtrid 2776 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2932  c0 4315   cint 4941   ciin 4989  Xcixp 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iin 4991  df-br 5140  df-opab 5202  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-ixp 8889
This theorem is referenced by: (None)
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