Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | tpstop.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps2 21537 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = ∪ 𝐽)) |
4 | 3 | simplbi 500 | 1 ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cuni 4831 ‘cfv 6349 Basecbs 16477 TopOpenctopn 16689 Topctop 21495 TopSpctps 21534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-top 21496 df-topon 21513 df-topsp 21535 |
This theorem is referenced by: mreclatdemoBAD 21698 prdstmdd 22726 invrcn 22783 cnextucn 22906 prdsxmslem2 23133 rlmbn 23958 sibfinima 31592 sibfof 31593 rrxtop 42568 |
Copyright terms: Public domain | W3C validator |