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Mirrors > Home > ILE Home > Th. List > uniixp | GIF version |
Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
uniixp | ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpf 6713 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | fssxp 5378 | . . . . 5 ⊢ (𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) |
4 | velpw 3581 | . . . 4 ⊢ (𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
5 | 3, 4 | sylibr 134 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) |
6 | 5 | ssriv 3159 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
7 | sspwuni 3968 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
8 | 6, 7 | mpbi 145 | 1 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 ⊆ wss 3129 𝒫 cpw 3574 ∪ cuni 3807 ∪ ciun 3884 × cxp 4620 ⟶wf 5207 Xcixp 6691 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ixp 6692 |
This theorem is referenced by: ixpexgg 6715 |
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