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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 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
| 2 | topnval.2 | . . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊) | |
| 3 | 1, 2 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
| 4 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
| 5 | topnval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
| 7 | 3, 6 | oveq12d 7449 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
| 8 | df-topn 17468 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
| 9 | ovex 7464 | . . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V | |
| 10 | 7, 8, 9 | fvmpt 7016 | . . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
| 11 | 10 | eqcomd 2743 | . 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
| 12 | 0rest 17474 | . . 3 ⊢ (∅ ↾t 𝐵) = ∅ | |
| 13 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅) | |
| 14 | 2, 13 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅) |
| 15 | 14 | oveq1d 7446 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵)) |
| 16 | fvprc 6898 | . . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅) | |
| 17 | 12, 15, 16 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
| 18 | 11, 17 | pm2.61i 182 | 1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TopSetcts 17303 ↾t crest 17465 TopOpenctopn 17466 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 df-topn 17468 |
| This theorem is referenced by: topnid 17480 topnpropd 17481 efmndtopn 18896 oppgtopn 19372 symgtopn 19424 mgptopn 20145 resstopn 23194 prdstopn 23636 tuslem 24275 tuslemOLD 24276 xrge0tsms 24856 om1opn 25069 xrge0tsmsd 33065 xrge0tmdALT 33945 |
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