Theorem ist1-3 23471

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 23469	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		toponmax 23048	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		eleq2 2858	. . . . . . . . 9 ⊢ (𝑜 = 𝑋 → (𝑥 ∈ 𝑜 ↔ 𝑥 ∈ 𝑋))
4	3	intminss 4940	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
5	2, 4	sylan 591	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
6	5	sselda 3945	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}) → 𝑦 ∈ 𝑋)
7		biimt 363	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
8	6, 7	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
9	8	ralbidva 3192	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥})))
10		id 23	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
11	10	rgenw 3089	. . . . . . . 8 ⊢ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
12		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	elintrab 4926	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜))
14	11, 13	mpbir 234	. . . . . . 7 ⊢ 𝑥 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}
15		snssi 4753	. . . . . . 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 3960	. . . . . 6 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}))
18	16, 17	mpbiran2 722	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥})
19		dfss3 3934	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
20	18, 19	bitri 278	. . . 4 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}𝑦 ∈ {𝑥})
21		vex 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elintrab 4926	. . . . . . 7 ⊢ (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
23		velsn 4607	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24		equcom 2045	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
25	23, 24	bitri 278	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
26	22, 25	imbi12i 353	. . . . . 6 ⊢ ((𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
27	26	ralbii 3117	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
28		ralcom3 3121	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} (𝑦 ∈ 𝑋 → 𝑦 ∈ {𝑥}))
29	27, 28	bitr3i 280	. . . 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 317	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
31	30	ralbidva 3192	. 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 285	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ⊆ wss 3913  {csn 4591  ∩ cint 4913  ‘cfv 6533  TopOnctopon 23032  Frect1 23429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541  df-topgen 17492  df-top 23016  df-topon 23033  df-cld 23141  df-t1 23436

This theorem is referenced by: (None)