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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 7374 | . 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾t (Base‘𝐾)) = ((TopSet‘𝐿) ↾t (Base‘𝐿))) |
4 | eqid 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | eqid 2736 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
6 | 4, 5 | topnval 17315 | . 2 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾) |
7 | eqid 2736 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
8 | eqid 2736 | . . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿) | |
9 | 7, 8 | topnval 17315 | . 2 ⊢ ((TopSet‘𝐿) ↾t (Base‘𝐿)) = (TopOpen‘𝐿) |
10 | 3, 6, 9 | 3eqtr3g 2799 | 1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 TopSetcts 17138 ↾t crest 17301 TopOpenctopn 17302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-rest 17303 df-topn 17304 |
This theorem is referenced by: sratopn 20651 tpsprop2d 22286 nrgtrg 24052 zhmnrg 32488 |
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