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Mirrors > Home > MPE Home > Th. List > topnpropd | Structured version Visualization version GIF version |
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
topnpropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
topnpropd.2 | ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) |
Ref | Expression |
---|---|
topnpropd | ⊢ (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 6810 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿))) |
4 | eqid 2771 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | eqid 2771 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
6 | 4, 5 | topnval 16302 | . 2 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾) |
7 | eqid 2771 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
8 | eqid 2771 | . . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿) | |
9 | 7, 8 | topnval 16302 | . 2 ⊢ ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿) |
10 | 3, 6, 9 | 3eqtr3g 2828 | 1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 TopSetcts 16154 ↾t crest 16288 TopOpenctopn 16289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-1st 7314 df-2nd 7315 df-rest 16290 df-topn 16291 |
This theorem is referenced by: sratopn 19399 tpsprop2d 20963 nrgtrg 22713 zhmnrg 30348 |
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