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Theorem ixpint 8174
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 8161 . . 3 (∀𝑥𝐴 𝐵 = 𝑦𝐵 𝑦X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦)
2 intiin 4763 . . . 4 𝐵 = 𝑦𝐵 𝑦
32a1i 11 . . 3 (𝑥𝐴 𝐵 = 𝑦𝐵 𝑦)
41, 3mprg 3106 . 2 X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦
5 ixpiin 8173 . 2 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝑦 = 𝑦𝐵 X𝑥𝐴 𝑦)
64, 5syl5eq 2844 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wne 2970  c0 4114   cint 4666   ciin 4710  Xcixp 8147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-nul 4982
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-int 4667  df-iin 4712  df-br 4843  df-opab 4905  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6063  df-fun 6102  df-fn 6103  df-fv 6108  df-ixp 8148
This theorem is referenced by: (None)
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