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Theorem ist1-3 21201
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 ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Distinct variable groups:   𝑥,𝑜,𝐽   𝑜,𝑋,𝑥

Proof of Theorem ist1-3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ist1-2 21199 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
2 toponmax 20778 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 eleq2 2719 . . . . . . . . 9 (𝑜 = 𝑋 → (𝑥𝑜𝑥𝑋))
43intminss 4535 . . . . . . . 8 ((𝑋𝐽𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
52, 4sylan 487 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
65sselda 3636 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → 𝑦𝑋)
7 biimt 349 . . . . . 6 (𝑦𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
98ralbidva 3014 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥})))
10 id 22 . . . . . . . . 9 (𝑥𝑜𝑥𝑜)
1110rgenw 2953 . . . . . . . 8 𝑜𝐽 (𝑥𝑜𝑥𝑜)
12 vex 3234 . . . . . . . . 9 𝑥 ∈ V
1312elintrab 4520 . . . . . . . 8 (𝑥 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑥𝑜))
1411, 13mpbir 221 . . . . . . 7 𝑥 {𝑜𝐽𝑥𝑜}
15 snssi 4371 . . . . . . 7 (𝑥 {𝑜𝐽𝑥𝑜} → {𝑥} ⊆ {𝑜𝐽𝑥𝑜})
1614, 15ax-mp 5 . . . . . 6 {𝑥} ⊆ {𝑜𝐽𝑥𝑜}
17 eqss 3651 . . . . . 6 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ {𝑜𝐽𝑥𝑜}))
1816, 17mpbiran2 974 . . . . 5 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ {𝑜𝐽𝑥𝑜} ⊆ {𝑥})
19 dfss3 3625 . . . . 5 ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
2018, 19bitri 264 . . . 4 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
21 vex 3234 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4520 . . . . . . 7 (𝑦 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
23 velsn 4226 . . . . . . . 8 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24 equcom 1991 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2523, 24bitri 264 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
2622, 25imbi12i 339 . . . . . 6 ((𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
2726ralbii 3009 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
28 ralcom3 3134 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
2927, 28bitr3i 266 . . . 4 (∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
309, 20, 293bitr4g 303 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
3130ralbidva 3014 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
321, 31bitr4d 271 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607  {csn 4210   cint 4507  cfv 5926  TopOnctopon 20763  Frect1 21159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-top 20747  df-topon 20764  df-cld 20871  df-t1 21166
This theorem is referenced by: (None)
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