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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 5906 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿))) |
4 | topnpropgd.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | eqid 2187 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2187 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
7 | 5, 6 | topnvalg 12717 | . . 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 2187 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
11 | eqid 2187 | . . . 4 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿) | |
12 | 10, 11 | topnvalg 12717 | . . 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 2228 | 1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 ‘cfv 5228 (class class class)co 5888 Basecbs 12475 TopSetcts 12556 ↾t crest 12705 TopOpenctopn 12706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-ndx 12478 df-slot 12479 df-base 12481 df-tset 12569 df-rest 12707 df-topn 12708 |
This theorem is referenced by: sratopng 13631 |
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