Theorem ixpint 8844

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 8830	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 → X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦)
2		intiin 5003	. . . 4 ⊢ ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦)
4	1, 3	mprg 3053	. 2 ⊢ X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦
5		ixpiin 8843	. 2 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)
6	4, 5	eqtrid 2778	1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∅c0 4278  ∩ cint 4892  ∩ ciin 4937  Xcixp 8816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iin 4939  df-br 5087  df-opab 5149  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484  df-ixp 8817

This theorem is referenced by: (None)