Theorem istps 22956

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 22955	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2831	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 22935	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		0ntop 22927	. . . . . 6 ⊢ ¬ ∅ ∈ Top
5		istps.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾)
6		fvprc 6899	. . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
7	5, 6	eqtrid 2787	. . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅)
8	7	eleq1d 2824	. . . . . 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 6907	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
13	12, 5	eqtr4di 2793	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14		fveq2 6907	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
16	14, 15	eqtr4di 2793	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
17	16	fveq2d 6911	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
18	13, 17	eleq12d 2833	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
19	11, 18	elab3 3689	. 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 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478  ∅c0 4339  ‘cfv 6563  Basecbs 17245  TopOpenctopn 17468  Topctop 22915  TopOnctopon 22932  TopSpctps 22954

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933  df-topsp 22955

This theorem is referenced by:  istps2  22957  tpspropd  22960  tsettps  22963  indistps2ALT  23038  resstps  23211  prdstps  23653  imastps  23745  xpstopnlem2  23835  tmdtopon  24105  tgptopon  24106  istgp2  24115  oppgtmd  24121  distgp  24123  indistgp  24124  efmndtmd  24125  qustgplem  24145  prdstmdd  24148  eltsms  24157  tsmscls  24162  tsmsgsum  24163  tsmsid  24164  tsmsmhm  24170  tsmsadd  24171  dvrcn  24208  cnmpt1vsca  24218  cnmpt2vsca  24219  tlmtgp  24220  ressusp  24289  tustps  24298  ucncn  24310  neipcfilu  24321  cnextucn  24328  ucnextcn  24329  isxms2  24474  ressxms  24554  prdsxmslem2  24558  nrgtrg  24727  cnfldtopon  24819  cnmpt1ds  24878  cnmpt2ds  24879  nmcn  24880  cnmpt1ip  25295  cnmpt2ip  25296  csscld  25297  clsocv  25298  minveclem4a  25478  rspectps  33844  mhmhmeotmd  33888  rrxtopon  46244  qndenserrnopnlem  46253