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Mirrors > Home > MPE Home > Th. List > topnval | Structured version Visualization version GIF version |
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
topnval.1 | ⊢ 𝐵 = (Base‘𝑊) |
topnval.2 | ⊢ 𝐽 = (TopSet‘𝑊) |
Ref | Expression |
---|---|
topnval | ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
2 | topnval.2 | . . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊) | |
3 | 1, 2 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
4 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | topnval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 4, 5 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
7 | 3, 6 | oveq12d 7163 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
8 | df-topn 16685 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
9 | ovex 7178 | . . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 6761 | . . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
11 | 10 | eqcomd 2824 | . 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
12 | 0rest 16691 | . . 3 ⊢ (∅ ↾t 𝐵) = ∅ | |
13 | fvprc 6656 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅) | |
14 | 2, 13 | syl5eq 2865 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅) |
15 | 14 | oveq1d 7160 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵)) |
16 | fvprc 6656 | . . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2879 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
18 | 11, 17 | pm2.61i 183 | 1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TopSetcts 16559 ↾t crest 16682 TopOpenctopn 16683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-rest 16684 df-topn 16685 |
This theorem is referenced by: topnid 16697 topnpropd 16698 oppgtopn 18419 symgtopn 18463 mgptopn 19177 resstopn 21722 prdstopn 22164 tuslem 22803 xrge0tsms 23369 om1opn 23567 xrge0tsmsd 30619 xrge0tmdALT 31088 efmndtopn 43981 |
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