Theorem istps 21536

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.

Step	Hyp	Ref	Expression
1		df-topsp 21535	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2904	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 21515	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		0ntop 21507	. . . . . 6 ⊢ ¬ ∅ ∈ Top
5		istps.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾)
6		fvprc 6657	. . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
7	5, 6	syl5eq 2868	. . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅)
8	7	eleq1d 2897	. . . . . 6 ⊢ (¬ 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top))
9	4, 8	mtbiri 329	. . . . 5 ⊢ (¬ 𝐾 ∈ V → ¬ 𝐽 ∈ Top)
10	9	con4i 114	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V)
11	3, 10	syl 17	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
12		fveq2 6664	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
13	12, 5	syl6eqr 2874	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14		fveq2 6664	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
16	14, 15	syl6eqr 2874	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
17	16	fveq2d 6668	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
18	13, 17	eleq12d 2907	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
19	11, 18	elab3 3673	. 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
20	2, 19	bitri 277	1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494  ∅c0 4290  ‘cfv 6349  Basecbs 16477  TopOpenctopn 16689  Topctop 21495  TopOnctopon 21512  TopSpctps 21534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-top 21496  df-topon 21513  df-topsp 21535

This theorem is referenced by:  istps2  21537  tpspropd  21540  tsettps  21543  indistps2ALT  21616  resstps  21789  prdstps  22231  imastps  22323  xpstopnlem2  22413  tmdtopon  22683  tgptopon  22684  istgp2  22693  oppgtmd  22699  distgp  22701  indistgp  22702  efmndtmd  22703  qustgplem  22723  prdstmdd  22726  eltsms  22735  tsmscls  22740  tsmsgsum  22741  tsmsid  22742  tsmsmhm  22748  tsmsadd  22749  dvrcn  22786  cnmpt1vsca  22796  cnmpt2vsca  22797  tlmtgp  22798  ressusp  22868  tustps  22876  ucncn  22888  neipcfilu  22899  cnextucn  22906  ucnextcn  22907  isxms2  23052  ressxms  23129  prdsxmslem2  23133  nrgtrg  23293  cnfldtopon  23385  cnmpt1ds  23444  cnmpt2ds  23445  nmcn  23446  cnmpt1ip  23844  cnmpt2ip  23845  csscld  23846  clsocv  23847  minveclem4a  24027  mhmhmeotmd  31165  rrxtopon  42567  qndenserrnopnlem  42576