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Mirrors > Home > MPE Home > Th. List > ixpint | Structured version Visualization version GIF version |
Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Ref | Expression |
---|---|
ixpint | ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq2 8902 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 → X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦) | |
2 | intiin 5053 | . . . 4 ⊢ ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦) |
4 | 1, 3 | mprg 3059 | . 2 ⊢ X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 |
5 | ixpiin 8915 | . 2 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | |
6 | 4, 5 | eqtrid 2776 | 1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 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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