Theorem istps 22940

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 22939	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2833	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 22919	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		0ntop 22911	. . . . . 6 ⊢ ¬ ∅ ∈ Top
5		istps.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾)
6		fvprc 6898	. . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
7	5, 6	eqtrid 2789	. . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅)
8	7	eleq1d 2826	. . . . . 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 6906	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
13	12, 5	eqtr4di 2795	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14		fveq2 6906	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
16	14, 15	eqtr4di 2795	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
17	16	fveq2d 6910	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
18	13, 17	eleq12d 2835	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
19	11, 18	elab3 3686	. 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 2714  Vcvv 3480  ∅c0 4333  ‘cfv 6561  Basecbs 17247  TopOpenctopn 17466  Topctop 22899  TopOnctopon 22916  TopSpctps 22938

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-topon 22917  df-topsp 22939

This theorem is referenced by:  istps2  22941  tpspropd  22944  tsettps  22947  indistps2ALT  23022  resstps  23195  prdstps  23637  imastps  23729  xpstopnlem2  23819  tmdtopon  24089  tgptopon  24090  istgp2  24099  oppgtmd  24105  distgp  24107  indistgp  24108  efmndtmd  24109  qustgplem  24129  prdstmdd  24132  eltsms  24141  tsmscls  24146  tsmsgsum  24147  tsmsid  24148  tsmsmhm  24154  tsmsadd  24155  dvrcn  24192  cnmpt1vsca  24202  cnmpt2vsca  24203  tlmtgp  24204  ressusp  24273  tustps  24282  ucncn  24294  neipcfilu  24305  cnextucn  24312  ucnextcn  24313  isxms2  24458  ressxms  24538  prdsxmslem2  24542  nrgtrg  24711  cnfldtopon  24803  cnmpt1ds  24864  cnmpt2ds  24865  nmcn  24866  cnmpt1ip  25281  cnmpt2ip  25282  csscld  25283  clsocv  25284  minveclem4a  25464  rspectps  33882  mhmhmeotmd  33926  rrxtopon  46303  qndenserrnopnlem  46312