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Theorem ist1-3 22684
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 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯}))
Distinct variable groups:   π‘₯,π‘œ,𝐽   π‘œ,𝑋,π‘₯

Proof of Theorem ist1-3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ist1-2 22682 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
2 toponmax 22259 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
3 eleq2 2826 . . . . . . . . 9 (π‘œ = 𝑋 β†’ (π‘₯ ∈ π‘œ ↔ π‘₯ ∈ 𝑋))
43intminss 4933 . . . . . . . 8 ((𝑋 ∈ 𝐽 ∧ π‘₯ ∈ 𝑋) β†’ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† 𝑋)
52, 4sylan 580 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† 𝑋)
65sselda 3942 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}) β†’ 𝑦 ∈ 𝑋)
7 biimt 360 . . . . . 6 (𝑦 ∈ 𝑋 β†’ (𝑦 ∈ {π‘₯} ↔ (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}) β†’ (𝑦 ∈ {π‘₯} ↔ (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
98ralbidva 3170 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
10 id 22 . . . . . . . . 9 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ)
1110rgenw 3066 . . . . . . . 8 βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ)
12 vex 3447 . . . . . . . . 9 π‘₯ ∈ V
1312elintrab 4919 . . . . . . . 8 (π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} ↔ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ))
1411, 13mpbir 230 . . . . . . 7 π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}
15 snssi 4766 . . . . . . 7 (π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ})
1614, 15ax-mp 5 . . . . . 6 {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}
17 eqss 3957 . . . . . 6 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯} ∧ {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}))
1816, 17mpbiran2 708 . . . . 5 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯})
19 dfss3 3930 . . . . 5 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯})
2018, 19bitri 274 . . . 4 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯})
21 vex 3447 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4919 . . . . . . 7 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} ↔ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ))
23 velsn 4600 . . . . . . . 8 (𝑦 ∈ {π‘₯} ↔ 𝑦 = π‘₯)
24 equcom 2021 . . . . . . . 8 (𝑦 = π‘₯ ↔ π‘₯ = 𝑦)
2523, 24bitri 274 . . . . . . 7 (𝑦 ∈ {π‘₯} ↔ π‘₯ = 𝑦)
2622, 25imbi12i 350 . . . . . 6 ((𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
2726ralbii 3094 . . . . 5 (βˆ€π‘¦ ∈ 𝑋 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
28 ralcom3 3098 . . . . 5 (βˆ€π‘¦ ∈ 𝑋 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯}))
2927, 28bitr3i 276 . . . 4 (βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯}))
309, 20, 293bitr4g 313 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
3130ralbidva 3170 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
321, 31bitr4d 281 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3062  {crab 3405   βŠ† wss 3908  {csn 4584  βˆ© cint 4905  β€˜cfv 6493  TopOnctopon 22243  Frect1 22642
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-topgen 17317  df-top 22227  df-topon 22244  df-cld 22354  df-t1 22649
This theorem is referenced by: (None)
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