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Mirrors > Home > MPE Home > Th. List > topnid | Structured version Visualization version GIF version |
Description: Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
topnval.1 | ⊢ 𝐵 = (Base‘𝑊) |
topnval.2 | ⊢ 𝐽 = (TopSet‘𝑊) |
Ref | Expression |
---|---|
topnid | ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topnval.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
2 | 1 | fvexi 6780 | . . 3 ⊢ 𝐵 ∈ V |
3 | restid2 17151 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾t 𝐵) = 𝐽) | |
4 | 2, 3 | mpan 687 | . 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾t 𝐵) = 𝐽) |
5 | topnval.2 | . . 3 ⊢ 𝐽 = (TopSet‘𝑊) | |
6 | 1, 5 | topnval 17155 | . 2 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
7 | 4, 6 | eqtr3di 2793 | 1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3429 ⊆ wss 3886 𝒫 cpw 4533 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 TopSetcts 16978 ↾t crest 17141 TopOpenctopn 17142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-rest 17143 df-topn 17144 |
This theorem is referenced by: topontopn 22099 prdstopn 22789 imastopn 22881 setsmstopn 23643 tngtopn 23824 circtopn 31795 rspectopn 31825 |
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