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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 2739 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | tpstop.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps2 21699 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = ∪ 𝐽)) |
4 | 3 | simplbi 501 | 1 ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∪ cuni 4806 ‘cfv 6350 Basecbs 16599 TopOpenctopn 16811 Topctop 21657 TopSpctps 21696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-iota 6308 df-fun 6352 df-fv 6358 df-top 21658 df-topon 21675 df-topsp 21697 |
This theorem is referenced by: mreclatdemoBAD 21860 prdstmdd 22888 invrcn 22945 cnextucn 23068 prdsxmslem2 23295 rlmbn 24126 sibfinima 31889 sibfof 31890 rrxtop 43413 |
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