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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 6645 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
2 | topnval.2 | . . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊) | |
3 | 1, 2 | eqtr4di 2851 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
4 | fveq2 6645 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | topnval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 4, 5 | eqtr4di 2851 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
7 | 3, 6 | oveq12d 7153 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
8 | df-topn 16689 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
9 | ovex 7168 | . . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 6745 | . . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
11 | 10 | eqcomd 2804 | . 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
12 | 0rest 16695 | . . 3 ⊢ (∅ ↾t 𝐵) = ∅ | |
13 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅) | |
14 | 2, 13 | syl5eq 2845 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅) |
15 | 14 | oveq1d 7150 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵)) |
16 | fvprc 6638 | . . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
18 | 11, 17 | pm2.61i 185 | 1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 TopSetcts 16563 ↾t crest 16686 TopOpenctopn 16687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-rest 16688 df-topn 16689 |
This theorem is referenced by: topnid 16701 topnpropd 16702 efmndtopn 18040 oppgtopn 18473 symgtopn 18526 mgptopn 19241 resstopn 21791 prdstopn 22233 tuslem 22873 xrge0tsms 23439 om1opn 23641 xrge0tsmsd 30742 xrge0tmdALT 31299 |
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