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Theorem topontopn 22797
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 22771 . . . 4 (𝐽 ∈ (TopOnβ€˜π΄) β†’ 𝐴 = βˆͺ 𝐽)
2 eqimss2 4036 . . . 4 (𝐴 = βˆͺ 𝐽 β†’ βˆͺ 𝐽 βŠ† 𝐴)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π΄) β†’ βˆͺ 𝐽 βŠ† 𝐴)
4 sspwuni 5096 . . 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 17390 . 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 3943  π’« cpw 4597  βˆͺ cuni 4902  β€˜cfv 6537  Basecbs 17153  TopSetcts 17212  TopOpenctopn 17376  TopOnctopon 22767
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-rest 17377  df-topn 17378  df-topon 22768
This theorem is referenced by:  tsettps  22798  xrstopn  23067  cnfldms  24647  cnfldtopn  24653
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