Theorem istps 14691

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 14690	. . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
2	1	eleq2i 2296	. 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3		topontop 14673	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4		topnfn 13263	. . . . . . 7 ⊢ TopOpen Fn V
5		fnrel 5415	. . . . . . 7 ⊢ (TopOpen Fn V → Rel TopOpen)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Rel TopOpen
7		0opn 14665	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
8		istps.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐾)
9	7, 8	eleqtrdi 2322	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾))
10		relelfvdm 5655	. . . . . 6 ⊢ ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen)
11	6, 9, 10	sylancr 414	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen)
12	11	elexd 2813	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V)
13	3, 12	syl 14	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
14		fveq2 5623	. . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
15	14, 8	eqtr4di 2280	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
16		fveq2 5623	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
17		istps.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐾)
18	16, 17	eqtr4di 2280	. . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
19	18	fveq2d 5627	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
20	15, 19	eleq12d 2300	. . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
21	13, 20	elab3 2955	. 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 1395   ∈ wcel 2200  {cab 2215  Vcvv 2799  ∅c0 3491  dom cdm 4716  Rel wrel 4721   Fn wfn 5309  ‘cfv 5314  Basecbs 13018  TopOpenctopn 13259  Topctop 14656  TopOnctopon 14669  TopSpctps 14689

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-ndx 13021  df-slot 13022  df-base 13024  df-tset 13115  df-rest 13260  df-topn 13261  df-top 14657  df-topon 14670  df-topsp 14690

This theorem is referenced by:  istps2  14692  tpspropd  14695  tsettps  14697  isxms2  15111  cnfldtopon  15199