Theorem ist1-3 23305

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 23303	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		toponmax 22882	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		eleq2 2826	. . . . . . . . 9 ⊢ (𝑜 = 𝑋 → (𝑥 ∈ 𝑜 ↔ 𝑥 ∈ 𝑋))
4	3	intminss 4931	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
5	2, 4	sylan 581	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
6	5	sselda 3935	. . . . . 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 3159	. . . 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 3056	. . . . . . . 8 ⊢ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
12		vex 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	elintrab 4917	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜))
14	11, 13	mpbir 231	. . . . . . 7 ⊢ 𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}
15		snssi 4766	. . . . . . 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 3951	. . . . . 6 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}))
18	16, 17	mpbiran2 711	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥})
19		dfss3 3924	. . . . 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 3446	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elintrab 4917	. . . . . . 7 ⊢ (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
23		velsn 4598	. . . . . . . 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 3084	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
28		ralcom3 3088	. . . . 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 3159	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  {csn 4582  ∩ cint 4904  ‘cfv 6500  TopOnctopon 22866  Frect1 23263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-topgen 17375  df-top 22850  df-topon 22867  df-cld 22975  df-t1 23270

This theorem is referenced by: (None)