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Theorem tsettps 12042
 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 12041 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4 id 19 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
53, 4eqeltrrd 2190 . 2 (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6 eqid 2113 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
71, 6istps 12036 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
85, 7sylibr 133 1 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1312   ∈ wcel 1461  ‘cfv 5079  Basecbs 11796  TopSetcts 11864  TopOpenctopn 11958  TopOnctopon 12014  TopSpctps 12034 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-5 8686  df-6 8687  df-7 8688  df-8 8689  df-9 8690  df-ndx 11799  df-slot 11800  df-base 11802  df-tset 11877  df-rest 11959  df-topn 11960  df-top 12002  df-topon 12015  df-topsp 12035 This theorem is referenced by:  eltpsg  12044
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