Theorem istps 14823

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 14822	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2298	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 14805	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		topnfn 13388	. . . . . . 7 ⊢ TopOpen Fn V
5		fnrel 5435	. . . . . . 7 ⊢ (TopOpen Fn V → Rel TopOpen)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Rel TopOpen
7		0opn 14797	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
8		istps.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐾)
9	7, 8	eleqtrdi 2324	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾))
10		relelfvdm 5680	. . . . . 6 ⊢ ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen)
11	6, 9, 10	sylancr 414	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen)
12	11	elexd 2817	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V)
13	3, 12	syl 14	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
14		fveq2 5648	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
15	14, 8	eqtr4di 2282	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
16		fveq2 5648	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
17		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
18	16, 17	eqtr4di 2282	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
19	18	fveq2d 5652	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
20	15, 19	eleq12d 2302	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
21	13, 20	elab3 2959	. 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 1398   ∈ wcel 2202  {cab 2217  Vcvv 2803  ∅c0 3496  dom cdm 4731  Rel wrel 4736   Fn wfn 5328  ‘cfv 5333  Basecbs 13143  TopOpenctopn 13384  Topctop 14788  TopOnctopon 14801  TopSpctps 14821

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-ndx 13146  df-slot 13147  df-base 13149  df-tset 13240  df-rest 13385  df-topn 13386  df-top 14789  df-topon 14802  df-topsp 14822

This theorem is referenced by:  istps2  14824  tpspropd  14827  tsettps  14829  isxms2  15243  cnfldtopon  15331