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Theorem ist1-3 21885
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 21883 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
2 toponmax 21462 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 eleq2 2898 . . . . . . . . 9 (𝑜 = 𝑋 → (𝑥𝑜𝑥𝑋))
43intminss 4893 . . . . . . . 8 ((𝑋𝐽𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
52, 4sylan 580 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
65sselda 3964 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → 𝑦𝑋)
7 biimt 362 . . . . . 6 (𝑦𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
98ralbidva 3193 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥})))
10 id 22 . . . . . . . . 9 (𝑥𝑜𝑥𝑜)
1110rgenw 3147 . . . . . . . 8 𝑜𝐽 (𝑥𝑜𝑥𝑜)
12 vex 3495 . . . . . . . . 9 𝑥 ∈ V
1312elintrab 4879 . . . . . . . 8 (𝑥 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑥𝑜))
1411, 13mpbir 232 . . . . . . 7 𝑥 {𝑜𝐽𝑥𝑜}
15 snssi 4733 . . . . . . 7 (𝑥 {𝑜𝐽𝑥𝑜} → {𝑥} ⊆ {𝑜𝐽𝑥𝑜})
1614, 15ax-mp 5 . . . . . 6 {𝑥} ⊆ {𝑜𝐽𝑥𝑜}
17 eqss 3979 . . . . . 6 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ {𝑜𝐽𝑥𝑜}))
1816, 17mpbiran2 706 . . . . 5 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ {𝑜𝐽𝑥𝑜} ⊆ {𝑥})
19 dfss3 3953 . . . . 5 ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
2018, 19bitri 276 . . . 4 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
21 vex 3495 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4879 . . . . . . 7 (𝑦 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
23 velsn 4573 . . . . . . . 8 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24 equcom 2016 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2523, 24bitri 276 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
2622, 25imbi12i 352 . . . . . 6 ((𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
2726ralbii 3162 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
28 ralcom3 3362 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
2927, 28bitr3i 278 . . . 4 (∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
309, 20, 293bitr4g 315 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
3130ralbidva 3193 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
321, 31bitr4d 283 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933  {csn 4557   cint 4867  cfv 6348  TopOnctopon 21446  Frect1 21843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-topgen 16705  df-top 21430  df-topon 21447  df-cld 21555  df-t1 21850
This theorem is referenced by: (None)
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