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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 6806 | . . 3 ⊢ 𝐵 ∈ V |
3 | restid2 17169 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾t 𝐵) = 𝐽) | |
4 | 2, 3 | mpan 686 | . 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾t 𝐵) = 𝐽) |
5 | topnval.2 | . . 3 ⊢ 𝐽 = (TopSet‘𝑊) | |
6 | 1, 5 | topnval 17173 | . 2 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
7 | 4, 6 | eqtr3di 2788 | 1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 Vcvv 3434 ⊆ wss 3889 𝒫 cpw 4536 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 TopSetcts 16996 ↾t crest 17159 TopOpenctopn 17160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-rest 17161 df-topn 17162 |
This theorem is referenced by: topontopn 22117 prdstopn 22807 imastopn 22899 setsmstopn 23661 tngtopn 23842 circtopn 31815 rspectopn 31845 |
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