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Theorem ist1-3 23373
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 23371 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
2 toponmax 22948 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 eleq2 2828 . . . . . . . . 9 (𝑜 = 𝑋 → (𝑥𝑜𝑥𝑋))
43intminss 4979 . . . . . . . 8 ((𝑋𝐽𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
52, 4sylan 580 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
65sselda 3995 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → 𝑦𝑋)
7 biimt 360 . . . . . 6 (𝑦𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
98ralbidva 3174 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥})))
10 id 22 . . . . . . . . 9 (𝑥𝑜𝑥𝑜)
1110rgenw 3063 . . . . . . . 8 𝑜𝐽 (𝑥𝑜𝑥𝑜)
12 vex 3482 . . . . . . . . 9 𝑥 ∈ V
1312elintrab 4965 . . . . . . . 8 (𝑥 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑥𝑜))
1411, 13mpbir 231 . . . . . . 7 𝑥 {𝑜𝐽𝑥𝑜}
15 snssi 4813 . . . . . . 7 (𝑥 {𝑜𝐽𝑥𝑜} → {𝑥} ⊆ {𝑜𝐽𝑥𝑜})
1614, 15ax-mp 5 . . . . . 6 {𝑥} ⊆ {𝑜𝐽𝑥𝑜}
17 eqss 4011 . . . . . 6 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ {𝑜𝐽𝑥𝑜}))
1816, 17mpbiran2 710 . . . . 5 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ {𝑜𝐽𝑥𝑜} ⊆ {𝑥})
19 dfss3 3984 . . . . 5 ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
2018, 19bitri 275 . . . 4 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
21 vex 3482 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4965 . . . . . . 7 (𝑦 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
23 velsn 4647 . . . . . . . 8 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24 equcom 2015 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2523, 24bitri 275 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
2622, 25imbi12i 350 . . . . . 6 ((𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
2726ralbii 3091 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
28 ralcom3 3095 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
2927, 28bitr3i 277 . . . 4 (∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
309, 20, 293bitr4g 314 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
3130ralbidva 3174 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
321, 31bitr4d 282 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963  {csn 4631   cint 4951  cfv 6563  TopOnctopon 22932  Frect1 23331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490  df-top 22916  df-topon 22933  df-cld 23043  df-t1 23338
This theorem is referenced by: (None)
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