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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 7158 | . . 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 6679 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4, 4 | xpex 7470 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
6 | 2, 5 | eqeltrrdi 2922 | . . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V) |
7 | uniexb 7480 | . . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V) | |
8 | 6, 7 | sylibr 236 | . . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V) |
9 | eqid 2821 | . . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈 | |
10 | 9 | restid 16701 | . . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈) |
12 | 1, 11 | eqtr2d 2857 | . 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵))) |
13 | ussval.2 | . . 3 ⊢ 𝑈 = (UnifSet‘𝑊) | |
14 | 3, 13 | ussval 22862 | . 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
15 | 12, 14 | syl6eq 2872 | 1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∪ cuni 4832 × cxp 5548 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 UnifSetcunif 16569 ↾t crest 16688 UnifStcuss 22856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-rest 16690 df-uss 22859 |
This theorem is referenced by: tususs 22873 cnflduss 23953 |
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