Theorem ist1-3 23578

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 23576	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
2		toponmax 23155	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		eleq2 2851	. . . . . . . . 9 ⊢ (𝑜 = 𝑋 → (𝑥 ∈ 𝑜 ↔ 𝑥 ∈ 𝑋))
4	3	intminss 4937	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
5	2, 4	sylan 592	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ 𝑋)
6	5	sselda 3934	. . . . . 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 3185	. . . 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 3082	. . . . . . . 8 ⊢ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑜)
12		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	elintrab 4923	. . . . . . . 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 4749	. . . . . . 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 3949	. . . . . 6 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜}))
18	16, 17	mpbiran2 723	. . . . 5 ⊢ (∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥} ↔ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ⊆ {𝑥})
19		dfss3 3923	. . . . 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 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elintrab 4923	. . . . . . 7 ⊢ (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} ↔ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
23		velsn 4603	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24		equcom 2051	. . . . . . . 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 3110	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 (𝑦 ∈ ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
28		ralcom3 3114	. . . . 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 3185	. 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}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ⊆ wss 3902  {csn 4587  ∩ cint 4910  ‘cfv 6537  TopOnctopon 23139  Frect1 23536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-topgen 17532  df-top 23123  df-topon 23140  df-cld 23248  df-t1 23543

This theorem is used by: (None)