Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6557 | . . 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 | fvex 6113 | . . . . . . . 8 ⊢ (Base‘𝑊) ∈ V | |
5 | 3, 4 | eqeltri 2684 | . . . . . . 7 ⊢ 𝐵 ∈ V |
6 | 5, 5 | xpex 6860 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
7 | 2, 6 | syl6eqelr 2697 | . . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V) |
8 | uniexb 6866 | . . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V) | |
9 | 7, 8 | sylibr 223 | . . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V) |
10 | eqid 2610 | . . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈 | |
11 | 10 | restid 15917 | . . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
13 | 1, 12 | eqtr2d 2645 | . 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵))) |
14 | ussval.2 | . . 3 ⊢ 𝑈 = (UnifSet‘𝑊) | |
15 | 3, 14 | ussval 21873 | . 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
16 | 13, 15 | syl6eq 2660 | 1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 UnifSetcunif 15778 ↾t crest 15904 UnifStcuss 21867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 df-uss 21870 |
This theorem is referenced by: tususs 21884 cnflduss 22960 |
Copyright terms: Public domain | W3C validator |