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Theorem topontopn 22946
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 22920 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = 𝐽)
2 eqimss2 4043 . . . 4 (𝐴 = 𝐽 𝐽𝐴)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽𝐴)
4 sspwuni 5100 . . 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 17480 . 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 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  cfv 6561  Basecbs 17247  TopSetcts 17303  TopOpenctopn 17466  TopOnctopon 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467  df-topn 17468  df-topon 22917
This theorem is referenced by:  tsettps  22947  xrstopn  23216  cnfldms  24796  cnfldtopn  24802
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