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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 2771 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | tpstop.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps2 20960 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = ∪ 𝐽)) |
4 | 3 | simplbi 485 | 1 ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∪ cuni 4574 ‘cfv 6031 Basecbs 16064 TopOpenctopn 16290 Topctop 20918 TopSpctps 20957 |
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-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 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-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 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-iota 5994 df-fun 6033 df-fv 6039 df-top 20919 df-topon 20936 df-topsp 20958 |
This theorem is referenced by: mreclatdemoBAD 21121 prdstmdd 22147 invrcn 22204 cnextucn 22327 prdsxmslem2 22554 rlmbn 23376 sibfinima 30741 sibfof 30742 rrxtop 41026 |
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