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Mirrors > Home > ILE Home > Th. List > ixpintm | GIF version |
Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.) |
Ref | Expression |
---|---|
ixpintm | ⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq2 6702 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 → X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦) | |
2 | intiin 3936 | . . . 4 ⊢ ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦 | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 𝑦) |
4 | 1, 3 | mprg 2532 | . 2 ⊢ X𝑥 ∈ 𝐴 ∩ 𝐵 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 |
5 | ixpiinm 6714 | . 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝑦 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | |
6 | 4, 5 | eqtrid 2220 | 1 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∃wex 1490 ∈ wcel 2146 ∩ cint 3840 ∩ ciin 3883 Xcixp 6688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iin 3885 df-br 3999 df-opab 4060 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 df-ixp 6689 |
This theorem is referenced by: (None) |
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