Theorem ixpint 8713

Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Assertion

Ref	Expression
ixpint	⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem ixpint

Step	Hyp	Ref	Expression
1		ixpeq2 8699	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 → X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦)
2		intiin 4989	. . . 4 ⊢ ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦)
4	1, 3	mprg 3078	. 2 ⊢ X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦
5		ixpiin 8712	. 2 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)
6	4, 5	eqtrid 2790	1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  ∩ cint 4879  ∩ ciin 4925  Xcixp 8685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iin 4927  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ixp 8686

This theorem is referenced by: (None)