Theorem ist1-3 23173

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 23171	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		toponmax 22748	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		eleq2 2821	. . . . . . . . 9 ⊢ (𝑜 = 𝑋 → (𝑥 ∈ 𝑜 ↔ 𝑥 ∈ 𝑋))
4	3	intminss 4978	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
5	2, 4	sylan 579	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
6	5	sselda 3982	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}) → 𝑦 ∈ 𝑋)
7		biimt 360	. . . . . 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 3174	. . . 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 3064	. . . . . . . 8 ⊢ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
12		vex 3477	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	elintrab 4964	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜))
14	11, 13	mpbir 230	. . . . . . 7 ⊢ 𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}
15		snssi 4811	. . . . . . 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 3997	. . . . . 6 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}))
18	16, 17	mpbiran2 707	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥})
19		dfss3 3970	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
20	18, 19	bitri 275	. . . 4 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
21		vex 3477	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elintrab 4964	. . . . . . 7 ⊢ (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
23		velsn 4644	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24		equcom 2020	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
25	23, 24	bitri 275	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
26	22, 25	imbi12i 350	. . . . . 6 ⊢ ((𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
27	26	ralbii 3092	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
28		ralcom3 3096	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥}))
29	27, 28	bitr3i 277	. . . 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 314	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
31	30	ralbidva 3174	. 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 282	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431   ⊆ wss 3948  {csn 4628  ∩ cint 4950  ‘cfv 6543  TopOnctopon 22732  Frect1 23131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17396  df-top 22716  df-topon 22733  df-cld 22843  df-t1 23138

This theorem is referenced by: (None)