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Mirrors > Home > MPE Home > Th. List > tpstop | Structured version Visualization version GIF version |
Description: The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.) |
Ref | Expression |
---|---|
tpstop.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
Ref | Expression |
---|---|
tpstop | ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | tpstop.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps2 22084 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = ∪ 𝐽)) |
4 | 3 | simplbi 498 | 1 ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∪ cuni 4839 ‘cfv 6433 Basecbs 16912 TopOpenctopn 17132 Topctop 22042 TopSpctps 22081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-top 22043 df-topon 22060 df-topsp 22082 |
This theorem is referenced by: mreclatdemoBAD 22247 prdstmdd 23275 invrcn 23332 cnextucn 23455 prdsxmslem2 23685 rlmbn 24525 sibfinima 32306 sibfof 32307 rrxtop 43830 |
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