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Mirrors > Home > MPE Home > Th. List > uniixp | Structured version Visualization version 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 8945 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | fssxp 6756 | . . . . 5 ⊢ (𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) |
4 | velpw 4611 | . . . 4 ⊢ (𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
5 | 3, 4 | sylibr 233 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) |
6 | 5 | ssriv 3986 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
7 | sspwuni 5107 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)) | |
8 | 6, 7 | mpbi 229 | 1 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3949 𝒫 cpw 4606 ∪ cuni 4912 ∪ ciun 5000 × cxp 5680 ⟶wf 6549 Xcixp 8922 |
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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ixp 8923 |
This theorem is referenced by: ixpexg 8947 |
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