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Theorem ist1-3 21957
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 21955 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
2 toponmax 21534 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 eleq2 2904 . . . . . . . . 9 (𝑜 = 𝑋 → (𝑥𝑜𝑥𝑋))
43intminss 4888 . . . . . . . 8 ((𝑋𝐽𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
52, 4sylan 583 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → {𝑜𝐽𝑥𝑜} ⊆ 𝑋)
65sselda 3953 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → 𝑦𝑋)
7 biimt 364 . . . . . 6 (𝑦𝑋 → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) ∧ 𝑦 {𝑜𝐽𝑥𝑜}) → (𝑦 ∈ {𝑥} ↔ (𝑦𝑋𝑦 ∈ {𝑥})))
98ralbidva 3191 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥})))
10 id 22 . . . . . . . . 9 (𝑥𝑜𝑥𝑜)
1110rgenw 3145 . . . . . . . 8 𝑜𝐽 (𝑥𝑜𝑥𝑜)
12 vex 3483 . . . . . . . . 9 𝑥 ∈ V
1312elintrab 4874 . . . . . . . 8 (𝑥 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑥𝑜))
1411, 13mpbir 234 . . . . . . 7 𝑥 {𝑜𝐽𝑥𝑜}
15 snssi 4725 . . . . . . 7 (𝑥 {𝑜𝐽𝑥𝑜} → {𝑥} ⊆ {𝑜𝐽𝑥𝑜})
1614, 15ax-mp 5 . . . . . 6 {𝑥} ⊆ {𝑜𝐽𝑥𝑜}
17 eqss 3968 . . . . . 6 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ∧ {𝑥} ⊆ {𝑜𝐽𝑥𝑜}))
1816, 17mpbiran2 709 . . . . 5 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ {𝑜𝐽𝑥𝑜} ⊆ {𝑥})
19 dfss3 3941 . . . . 5 ( {𝑜𝐽𝑥𝑜} ⊆ {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
2018, 19bitri 278 . . . 4 ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦 {𝑜𝐽𝑥𝑜}𝑦 ∈ {𝑥})
21 vex 3483 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4874 . . . . . . 7 (𝑦 {𝑜𝐽𝑥𝑜} ↔ ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
23 velsn 4566 . . . . . . . 8 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
24 equcom 2026 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2523, 24bitri 278 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑥 = 𝑦)
2622, 25imbi12i 354 . . . . . 6 ((𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
2726ralbii 3160 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
28 ralcom3 3355 . . . . 5 (∀𝑦𝑋 (𝑦 {𝑜𝐽𝑥𝑜} → 𝑦 ∈ {𝑥}) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
2927, 28bitr3i 280 . . . 4 (∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ ∀𝑦 {𝑜𝐽𝑥𝑜} (𝑦𝑋𝑦 ∈ {𝑥}))
309, 20, 293bitr4g 317 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ( {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
3130ralbidva 3191 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥} ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
321, 31bitr4d 285 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  wss 3919  {csn 4550   cint 4862  cfv 6343  TopOnctopon 21518  Frect1 21915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-topgen 16717  df-top 21502  df-topon 21519  df-cld 21627  df-t1 21922
This theorem is referenced by: (None)
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