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Theorem tsettps 21641
 Description: If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tsettps.a 𝐴 = (Base‘𝐾)
tsettps.j 𝐽 = (TopSet‘𝐾)
Assertion
Ref Expression
tsettps (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Proof of Theorem tsettps
StepHypRef Expression
1 tsettps.a . . . 4 𝐴 = (Base‘𝐾)
2 tsettps.j . . . 4 𝐽 = (TopSet‘𝐾)
31, 2topontopn 21640 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4 id 22 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
53, 4eqeltrrd 2853 . 2 (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6 eqid 2758 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
71, 6istps 21634 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
85, 7sylibr 237 1 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  Basecbs 16541  TopSetcts 16629  TopOpenctopn 16753  TopOnctopon 21610  TopSpctps 21632 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-rest 16754  df-topn 16755  df-top 21594  df-topon 21611  df-topsp 21633 This theorem is referenced by:  eltpsg  21643  indistpsALT  21713  xrstps  21909  prdstps  22329
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