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Mirrors > Home > MPE Home > Th. List > ustuni | Structured version Visualization version GIF version |
Description: The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
Ref | Expression |
---|---|
ustuni | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustbasel 23486 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈) | |
2 | ustssxp 23484 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋)) | |
3 | 2 | ralrimiva 3142 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋)) |
4 | pwssb 5060 | . . 3 ⊢ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋)) | |
5 | 3, 4 | sylibr 233 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) |
6 | elpwuni 5064 | . . 3 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∪ 𝑈 = (𝑋 × 𝑋))) | |
7 | 6 | biimpa 478 | . 2 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → ∪ 𝑈 = (𝑋 × 𝑋)) |
8 | 1, 5, 7 | syl2anc 585 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ⊆ wss 3909 𝒫 cpw 4559 ∪ cuni 4864 × cxp 5629 ‘cfv 6492 UnifOncust 23479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-res 5643 df-iota 6444 df-fun 6494 df-fv 6500 df-ust 23480 |
This theorem is referenced by: tususs 23550 cnflduss 24648 |
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