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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 7422 | . . 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 6905 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4, 4 | xpex 7749 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
6 | 2, 5 | eqeltrrdi 2837 | . . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V) |
7 | uniexb 7760 | . . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V) | |
8 | 6, 7 | sylibr 233 | . . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V) |
9 | eqid 2727 | . . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈 | |
10 | 9 | restid 17408 | . . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
12 | 1, 11 | eqtr2d 2768 | . 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵))) |
13 | ussval.2 | . . 3 ⊢ 𝑈 = (UnifSet‘𝑊) | |
14 | 3, 13 | ussval 24157 | . 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
15 | 12, 14 | eqtrdi 2783 | 1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cuni 4903 × cxp 5670 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 UnifSetcunif 17236 ↾t crest 17395 UnifStcuss 24151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-rest 17397 df-uss 24154 |
This theorem is referenced by: tususs 24168 cnflduss 25277 |
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