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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 6930 | . . 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 6460 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4, 4 | xpex 7240 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
6 | 2, 5 | syl6eqelr 2867 | . . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V) |
7 | uniexb 7250 | . . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V) | |
8 | 6, 7 | sylibr 226 | . . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V) |
9 | eqid 2777 | . . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈 | |
10 | 9 | restid 16480 | . . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
12 | 1, 11 | eqtr2d 2814 | . 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵))) |
13 | ussval.2 | . . 3 ⊢ 𝑈 = (UnifSet‘𝑊) | |
14 | 3, 13 | ussval 22471 | . 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
15 | 12, 14 | syl6eq 2829 | 1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∪ cuni 4671 × cxp 5353 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 UnifSetcunif 16348 ↾t crest 16467 UnifStcuss 22465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-rest 16469 df-uss 22468 |
This theorem is referenced by: tususs 22482 cnflduss 23562 |
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