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Theorem ist1-3 20963
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 20961 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
2 toponmax 20543 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 eleq2 2677 . . . . . . . . 9 (𝑜 = 𝑋 → (𝑥𝑜𝑥𝑋))
43intminss 4438 . . . . . . . 8 ((𝑋𝐽𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
52, 4sylan 487 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
65sselda 3568 . . . . . 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 2968 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥})))
10 id 22 . . . . . . . . 9 (𝑥𝑜𝑥𝑜)
1110rgenw 2908 . . . . . . . 8 𝑜𝐽 (𝑥𝑜𝑥𝑜)
12 vex 3176 . . . . . . . . 9 𝑥 ∈ V
1312elintrab 4423 . . . . . . . 8 (𝑥 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑥𝑜))
1411, 13mpbir 220 . . . . . . 7 𝑥 {𝑜𝐽𝑥𝑜}
15 snssi 4280 . . . . . . 7 (𝑥 {𝑜𝐽𝑥𝑜} → {𝑥} ⊆ {𝑜𝐽𝑥𝑜})
1614, 15ax-mp 5 . . . . . 6 {𝑥} ⊆ {𝑜𝐽𝑥𝑜}
17 eqss 3583 . . . . . 6 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ {𝑜𝐽𝑥𝑜}))
1816, 17mpbiran2 956 . . . . 5 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ {𝑜𝐽𝑥𝑜} ⊆ {𝑥})
19 dfss3 3558 . . . . 5 ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
2018, 19bitri 263 . . . 4 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
21 vex 3176 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4423 . . . . . . 7 (𝑦 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
23 velsn 4141 . . . . . . . 8 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24 equcom 1932 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2523, 24bitri 263 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
2622, 25imbi12i 339 . . . . . 6 ((𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
2726ralbii 2963 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
28 ralcom3 3084 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
2927, 28bitr3i 265 . . . 4 (∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
309, 20, 293bitr4g 302 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
3130ralbidva 2968 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
321, 31bitr4d 270 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  wss 3540  {csn 4125   cint 4410  cfv 5804  TopOnctopon 20518  Frect1 20921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-top 20521  df-topon 20523  df-cld 20633  df-t1 20928
This theorem is referenced by: (None)
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