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Theorem tsettps 22196
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 22195 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
4 id 22 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ (TopOn‘𝐴))
53, 4eqeltrrd 2838 . 2 (𝐽 ∈ (TopOn‘𝐴) → (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
6 eqid 2736 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
71, 6istps 22189 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘𝐴))
85, 7sylibr 233 1 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cfv 6479  Basecbs 17009  TopSetcts 17065  TopOpenctopn 17229  TopOnctopon 22165  TopSpctps 22187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-rest 17230  df-topn 17231  df-top 22149  df-topon 22166  df-topsp 22188
This theorem is referenced by:  eltpsg  22198  eltpsgOLD  22199  indistpsALT  22269  indistpsALTOLD  22270  xrstps  22466  prdstps  22886
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