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Theorem istps 20678
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Base‘𝐾)
istps.j 𝐽 = (TopOpen‘𝐾)
Assertion
Ref Expression
istps (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Proof of Theorem istps
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df-topsp 20677 . . 3 TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
21eleq2i 2690 . 2 (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3 topontop 20658 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4 0ntop 20650 . . . . . 6 ¬ ∅ ∈ Top
5 istps.j . . . . . . . 8 𝐽 = (TopOpen‘𝐾)
6 fvprc 6152 . . . . . . . 8 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
75, 6syl5eq 2667 . . . . . . 7 𝐾 ∈ V → 𝐽 = ∅)
87eleq1d 2683 . . . . . 6 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top))
94, 8mtbiri 317 . . . . 5 𝐾 ∈ V → ¬ 𝐽 ∈ Top)
109con4i 113 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ V)
113, 10syl 17 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
12 fveq2 6158 . . . . 5 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
1312, 5syl6eqr 2673 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14 fveq2 6158 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15 istps.a . . . . . 6 𝐴 = (Base‘𝐾)
1614, 15syl6eqr 2673 . . . . 5 (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
1716fveq2d 6162 . . . 4 (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
1813, 17eleq12d 2692 . . 3 (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
1911, 18elab3 3346 . 2 (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
202, 19bitri 264 1 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3190  c0 3897  cfv 5857  Basecbs 15800  TopOpenctopn 16022  Topctop 20638  TopOnctopon 20655  TopSpctps 20676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-top 20639  df-topon 20656  df-topsp 20677
This theorem is referenced by:  istps2  20679  tpspropd  20682  tsettps  20685  indistps2ALT  20758  resstps  20931  prdstps  21372  imastps  21464  xpstopnlem2  21554  tmdtopon  21825  tgptopon  21826  istgp2  21835  oppgtmd  21841  distgp  21843  indistgp  21844  symgtgp  21845  qustgplem  21864  prdstmdd  21867  eltsms  21876  tsmscls  21881  tsmsgsum  21882  tsmsid  21883  tsmsmhm  21889  tsmsadd  21890  dvrcn  21927  cnmpt1vsca  21937  cnmpt2vsca  21938  tlmtgp  21939  ressusp  22009  tustps  22017  ucncn  22029  neipcfilu  22040  cnextucn  22047  ucnextcn  22048  isxms2  22193  ressxms  22270  prdsxmslem2  22274  nrgtrg  22434  cnfldtopon  22526  cnmpt1ds  22585  cnmpt2ds  22586  nmcn  22587  cnmpt1ip  22986  cnmpt2ip  22987  csscld  22988  clsocv  22989  minveclem4a  23141  mhmhmeotmd  29797  rrxtopon  39845  qndenserrnopnlem  39854
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