Theorem ist1-3 23332

Description: A space is T₁ iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ist1-3	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥}))

Distinct variable groups:   𝑥,𝑜,𝐽   𝑜,𝑋,𝑥

Proof of Theorem ist1-3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ist1-2 23330	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		toponmax 22909	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		eleq2 2828	. . . . . . . . 9 ⊢ (𝑜 = 𝑋 → (𝑥 ∈ 𝑜 ↔ 𝑥 ∈ 𝑋))
4	3	intminss 4904	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
5	2, 4	sylan 586	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
6	5	sselda 3915	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}) → 𝑦 ∈ 𝑋)
7		biimt 361	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
8	6, 7	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
9	8	ralbidva 3160	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
10		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
11	10	rgenw 3057	. . . . . . . 8 ⊢ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
12		vex 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	elintrab 4890	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜))
14	11, 13	mpbir 232	. . . . . . 7 ⊢ 𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}
15		snssi 4717	. . . . . . 7 ⊢ (𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜})
16	14, 15	ax-mp 5	. . . . . 6 ⊢ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}
17		eqss 3930	. . . . . 6 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}))
18	16, 17	mpbiran2 716	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥})
19		dfss3 3904	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
20	18, 19	bitri 276	. . . 4 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
21		vex 3435	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elintrab 4890	. . . . . . 7 ⊢ (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
23		velsn 4571	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24		equcom 2025	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
25	23, 24	bitri 276	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
26	22, 25	imbi12i 351	. . . . . 6 ⊢ ((𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
27	26	ralbii 3085	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
28		ralcom3 3089	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥}))
29	27, 28	bitr3i 278	. . . 4 ⊢ (∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥}))
30	9, 20, 29	3bitr4g 315	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
31	30	ralbidva 3160	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
32	1, 31	bitr4d 283	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   ⊆ wss 3883  {csn 4555  ∩ cint 4877  ‘cfv 6485  TopOnctopon 22893  Frect1 23290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-topgen 17397  df-top 22877  df-topon 22894  df-cld 23002  df-t1 23297

This theorem is referenced by: (None)