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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 22815 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈) | |
2 | ustssxp 22813 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋)) | |
3 | 2 | ralrimiva 3182 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋)) |
4 | pwssb 5023 | . . 3 ⊢ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋)) | |
5 | 3, 4 | sylibr 236 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) |
6 | elpwuni 5027 | . . 3 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∪ 𝑈 = (𝑋 × 𝑋))) | |
7 | 6 | biimpa 479 | . 2 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → ∪ 𝑈 = (𝑋 × 𝑋)) |
8 | 1, 5, 7 | syl2anc 586 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 × cxp 5553 ‘cfv 6355 UnifOncust 22808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-ust 22809 |
This theorem is referenced by: tususs 22879 cnflduss 23959 |
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