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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 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
2 | topnval.2 | . . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊) | |
3 | 1, 2 | syl6eqr 2662 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | topnval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 4, 5 | syl6eqr 2662 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
7 | 3, 6 | oveq12d 6567 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
8 | df-topn 15907 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
9 | ovex 6577 | . . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V | |
10 | 7, 8, 9 | fvmpt 6191 | . . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
11 | 10 | eqcomd 2616 | . 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
12 | 0rest 15913 | . . 3 ⊢ (∅ ↾t 𝐵) = ∅ | |
13 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅) | |
14 | 2, 13 | syl5eq 2656 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅) |
15 | 14 | oveq1d 6564 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵)) |
16 | fvprc 6097 | . . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅) | |
17 | 12, 15, 16 | 3eqtr4a 2670 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
18 | 11, 17 | pm2.61i 175 | 1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TopSetcts 15774 ↾t crest 15904 TopOpenctopn 15905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 df-topn 15907 |
This theorem is referenced by: topnid 15919 topnpropd 15920 oppgtopn 17606 symgtopn 17648 mgptopn 18321 resstopn 20800 prdstopn 21241 tuslem 21881 xrge0tsms 22445 om1opn 22644 xrge0tsmsd 29116 xrge0tmdOLD 29319 |
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