Theorem istps 22436

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 22435	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2826	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 22415	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		0ntop 22407	. . . . . 6 ⊢ ¬ ∅ ∈ Top
5		istps.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾)
6		fvprc 6884	. . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅)
7	5, 6	eqtrid 2785	. . . . . . 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 6892	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
13	12, 5	eqtr4di 2791	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
14		fveq2 6892	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
15		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
16	14, 15	eqtr4di 2791	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
17	16	fveq2d 6896	. . . 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 3677	. 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 205   = wceq 1542   ∈ wcel 2107  {cab 2710  Vcvv 3475  ∅c0 4323  ‘cfv 6544  Basecbs 17144  TopOpenctopn 17367  Topctop 22395  TopOnctopon 22412  TopSpctps 22434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-top 22396  df-topon 22413  df-topsp 22435

This theorem is referenced by:  istps2  22437  tpspropd  22440  tsettps  22443  indistps2ALT  22518  resstps  22691  prdstps  23133  imastps  23225  xpstopnlem2  23315  tmdtopon  23585  tgptopon  23586  istgp2  23595  oppgtmd  23601  distgp  23603  indistgp  23604  efmndtmd  23605  qustgplem  23625  prdstmdd  23628  eltsms  23637  tsmscls  23642  tsmsgsum  23643  tsmsid  23644  tsmsmhm  23650  tsmsadd  23651  dvrcn  23688  cnmpt1vsca  23698  cnmpt2vsca  23699  tlmtgp  23700  ressusp  23769  tustps  23778  ucncn  23790  neipcfilu  23801  cnextucn  23808  ucnextcn  23809  isxms2  23954  ressxms  24034  prdsxmslem2  24038  nrgtrg  24207  cnfldtopon  24299  cnmpt1ds  24358  cnmpt2ds  24359  nmcn  24360  cnmpt1ip  24764  cnmpt2ip  24765  csscld  24766  clsocv  24767  minveclem4a  24947  rspectps  32863  mhmhmeotmd  32907  rrxtopon  45004  qndenserrnopnlem  45013