Theorem istps 22872

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 22871	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2826	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 22851	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		0ntop 22843	. . . . . 6 ⊢ ¬ ∅ ∈ Top
5		istps.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾)
6		fvprc 6868	. . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
7	5, 6	eqtrid 2782	. . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅)
8	7	eleq1d 2819	. . . . . 6 ⊢ (¬ 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top))
9	4, 8	mtbiri 327	. . . . 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 6876	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
13	12, 5	eqtr4di 2788	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14		fveq2 6876	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
16	14, 15	eqtr4di 2788	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
17	16	fveq2d 6880	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
18	13, 17	eleq12d 2828	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
19	11, 18	elab3 3665	. 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
20	2, 19	bitri 275	1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {cab 2713  Vcvv 3459  ∅c0 4308  ‘cfv 6531  Basecbs 17228  TopOpenctopn 17435  Topctop 22831  TopOnctopon 22848  TopSpctps 22870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-top 22832  df-topon 22849  df-topsp 22871

This theorem is referenced by:  istps2  22873  tpspropd  22876  tsettps  22879  indistps2ALT  22952  resstps  23125  prdstps  23567  imastps  23659  xpstopnlem2  23749  tmdtopon  24019  tgptopon  24020  istgp2  24029  oppgtmd  24035  distgp  24037  indistgp  24038  efmndtmd  24039  qustgplem  24059  prdstmdd  24062  eltsms  24071  tsmscls  24076  tsmsgsum  24077  tsmsid  24078  tsmsmhm  24084  tsmsadd  24085  dvrcn  24122  cnmpt1vsca  24132  cnmpt2vsca  24133  tlmtgp  24134  ressusp  24203  tustps  24211  ucncn  24223  neipcfilu  24234  cnextucn  24241  ucnextcn  24242  isxms2  24387  ressxms  24464  prdsxmslem2  24468  nrgtrg  24629  cnfldtopon  24721  cnmpt1ds  24782  cnmpt2ds  24783  nmcn  24784  cnmpt1ip  25199  cnmpt2ip  25200  csscld  25201  clsocv  25202  minveclem4a  25382  rspectps  33914  mhmhmeotmd  33958  rrxtopon  46317  qndenserrnopnlem  46326