Theorem istps 14211

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 14210	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2260	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 14193	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		topnfn 12858	. . . . . . 7 ⊢ TopOpen Fn V
5		fnrel 5353	. . . . . . 7 ⊢ (TopOpen Fn V → Rel TopOpen)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Rel TopOpen
7		0opn 14185	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
8		istps.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐾)
9	7, 8	eleqtrdi 2286	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾))
10		relelfvdm 5587	. . . . . 6 ⊢ ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen)
11	6, 9, 10	sylancr 414	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen)
12	11	elexd 2773	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V)
13	3, 12	syl 14	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
14		fveq2 5555	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
15	14, 8	eqtr4di 2244	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
16		fveq2 5555	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
17		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
18	16, 17	eqtr4di 2244	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
19	18	fveq2d 5559	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
20	15, 19	eleq12d 2264	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
21	13, 20	elab3 2913	. 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
22	2, 21	bitri 184	1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760  ∅c0 3447  dom cdm 4660  Rel wrel 4665   Fn wfn 5250  ‘cfv 5255  Basecbs 12621  TopOpenctopn 12854  Topctop 14176  TopOnctopon 14189  TopSpctps 14209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-ndx 12624  df-slot 12625  df-base 12627  df-tset 12717  df-rest 12855  df-topn 12856  df-top 14177  df-topon 14190  df-topsp 14210

This theorem is referenced by:  istps2  14212  tpspropd  14215  tsettps  14217  isxms2  14631  cnfldtopon  14719