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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 8458 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 → X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦) | |
2 | intiin 4946 | . . . 4 ⊢ ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦) |
4 | 1, 3 | mprg 3120 | . 2 ⊢ X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 |
5 | ixpiin 8471 | . 2 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | |
6 | 4, 5 | syl5eq 2845 | 1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ∩ cint 4838 ∩ ciin 4882 Xcixp 8444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iin 4884 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ixp 8445 |
This theorem is referenced by: (None) |
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