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Mirrors > Home > ILE Home > Th. List > topnpropgd | GIF version |
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.) |
Ref | Expression |
---|---|
topnpropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
topnpropd.2 | ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) |
topnpropgd.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
topnpropgd.l | ⊢ (𝜑 → 𝐿 ∈ 𝑊) |
Ref | Expression |
---|---|
topnpropgd | ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topnpropd.2 | . . 3 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) | |
2 | topnpropd.1 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
3 | 1, 2 | oveq12d 5854 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿))) |
4 | topnpropgd.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | eqid 2164 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2164 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
7 | 5, 6 | topnvalg 12510 | . . 3 ⊢ (𝐾 ∈ 𝑉 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)) |
8 | 4, 7 | syl 14 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)) |
9 | topnpropgd.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ 𝑊) | |
10 | eqid 2164 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
11 | eqid 2164 | . . . 4 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿) | |
12 | 10, 11 | topnvalg 12510 | . . 3 ⊢ (𝐿 ∈ 𝑊 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)) |
13 | 9, 12 | syl 14 | . 2 ⊢ (𝜑 → ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿)) |
14 | 3, 8, 13 | 3eqtr3d 2205 | 1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ‘cfv 5182 (class class class)co 5836 Basecbs 12337 TopSetcts 12405 ↾t crest 12498 TopOpenctopn 12499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 |
This theorem is referenced by: (None) |
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