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Theorem topontopn 21252
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 21226 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = 𝐽)
2 eqimss2 3914 . . . 4 (𝐴 = 𝐽 𝐽𝐴)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐽𝐴)
4 sspwuni 4888 . . 3 (𝐽 ⊆ 𝒫 𝐴 𝐽𝐴)
53, 4sylibr 226 . 2 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ⊆ 𝒫 𝐴)
6 tsettps.a . . 3 𝐴 = (Base‘𝐾)
7 tsettps.j . . 3 𝐽 = (TopSet‘𝐾)
86, 7topnid 16565 . 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 1507  wcel 2050  wss 3829  𝒫 cpw 4422   cuni 4712  cfv 6188  Basecbs 16339  TopSetcts 16427  TopOpenctopn 16551  TopOnctopon 21222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-rest 16552  df-topn 16553  df-topon 21223
This theorem is referenced by:  tsettps  21253  xrstopn  21520  cnfldms  23087  cnfldtopn  23093
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