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Theorem topontopn 22870
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tsettps.a 𝐴 = (Baseβ€˜πΎ)
tsettps.j 𝐽 = (TopSetβ€˜πΎ)
Assertion
Ref Expression
topontopn (𝐽 ∈ (TopOnβ€˜π΄) β†’ 𝐽 = (TopOpenβ€˜πΎ))

Proof of Theorem topontopn
StepHypRef Expression
1 toponuni 22844 . . . 4 (𝐽 ∈ (TopOnβ€˜π΄) β†’ 𝐴 = βˆͺ 𝐽)
2 eqimss2 4041 . . . 4 (𝐴 = βˆͺ 𝐽 β†’ βˆͺ 𝐽 βŠ† 𝐴)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π΄) β†’ βˆͺ 𝐽 βŠ† 𝐴)
4 sspwuni 5107 . . 3 (𝐽 βŠ† 𝒫 𝐴 ↔ βˆͺ 𝐽 βŠ† 𝐴)
53, 4sylibr 233 . 2 (𝐽 ∈ (TopOnβ€˜π΄) β†’ 𝐽 βŠ† 𝒫 𝐴)
6 tsettps.a . . 3 𝐴 = (Baseβ€˜πΎ)
7 tsettps.j . . 3 𝐽 = (TopSetβ€˜πΎ)
86, 7topnid 17426 . 2 (𝐽 βŠ† 𝒫 𝐴 β†’ 𝐽 = (TopOpenβ€˜πΎ))
95, 8syl 17 1 (𝐽 ∈ (TopOnβ€˜π΄) β†’ 𝐽 = (TopOpenβ€˜πΎ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  π’« cpw 4606  βˆͺ cuni 4912  β€˜cfv 6553  Basecbs 17189  TopSetcts 17248  TopOpenctopn 17412  TopOnctopon 22840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-rest 17413  df-topn 17414  df-topon 22841
This theorem is referenced by:  tsettps  22871  xrstopn  23140  cnfldms  24720  cnfldtopn  24726
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