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Theorem topontopn 22962
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tsettps.a 𝐴 = (Base‘𝐾)
tsettps.j 𝐽 = (TopSet‘𝐾)
Assertion
Ref Expression
topontopn (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))

Proof of Theorem topontopn
StepHypRef Expression
1 toponuni 22936 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = 𝐽)
2 eqimss2 4055 . . . 4 (𝐴 = 𝐽 𝐽𝐴)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽𝐴)
4 sspwuni 5105 . . 3 (𝐽 ⊆ 𝒫 𝐴 𝐽𝐴)
53, 4sylibr 234 . 2 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ⊆ 𝒫 𝐴)
6 tsettps.a . . 3 𝐴 = (Base‘𝐾)
7 tsettps.j . . 3 𝐽 = (TopSet‘𝐾)
86, 7topnid 17482 . 2 (𝐽 ⊆ 𝒫 𝐴𝐽 = (TopOpen‘𝐾))
95, 8syl 17 1 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605   cuni 4912  cfv 6563  Basecbs 17245  TopSetcts 17304  TopOpenctopn 17468  TopOnctopon 22932
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-topn 17470  df-topon 22933
This theorem is referenced by:  tsettps  22963  xrstopn  23232  cnfldms  24812  cnfldtopn  24818
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