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| Mirrors > Home > MPE Home > Th. List > ussid | Structured version Visualization version GIF version | ||
| Description: In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.) |
| Ref | Expression |
|---|---|
| ussval.1 | ⊢ 𝐵 = (Base‘𝑊) |
| ussval.2 | ⊢ 𝑈 = (UnifSet‘𝑊) |
| Ref | Expression |
|---|---|
| ussid | ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 7439 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t (𝐵 × 𝐵)) = (𝑈 ↾t ∪ 𝑈)) | |
| 2 | id 22 | . . . . . 6 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝐵 × 𝐵) = ∪ 𝑈) | |
| 3 | ussval.1 | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊) | |
| 4 | 3 | fvexi 6920 | . . . . . . 7 ⊢ 𝐵 ∈ V |
| 5 | 4, 4 | xpex 7773 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
| 6 | 2, 5 | eqeltrrdi 2850 | . . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V) |
| 7 | uniexb 7784 | . . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V) | |
| 8 | 6, 7 | sylibr 234 | . . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V) |
| 9 | eqid 2737 | . . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈 | |
| 10 | 9 | restid 17478 | . . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
| 11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
| 12 | 1, 11 | eqtr2d 2778 | . 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵))) |
| 13 | ussval.2 | . . 3 ⊢ 𝑈 = (UnifSet‘𝑊) | |
| 14 | 3, 13 | ussval 24268 | . 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
| 15 | 12, 14 | eqtrdi 2793 | 1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 UnifSetcunif 17307 ↾t crest 17465 UnifStcuss 24262 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 df-uss 24265 |
| This theorem is referenced by: tususs 24279 cnflduss 25390 |
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