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Theorem ist1-2 22851
Description: An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ist1-2 (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)))
Distinct variable groups:   π‘₯,𝑦,π‘œ,𝐽   π‘œ,𝑋,π‘₯,𝑦
Proof of Theorem ist1-2
StepHypRef Expression
1 topontop 22415 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 eqid 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
32ist1 22825 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
43baib 537 . . 3 (𝐽 ∈ Top β†’ (𝐽 ∈ Fre ↔ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
51, 4syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
6 toponuni 22416 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76raleqdv 3326 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘¦ ∈ 𝑋 {𝑦} ∈ (Clsdβ€˜π½) ↔ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
81adantr 482 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝐽 ∈ Top)
9 eltop2 22478 . . . . . 6 (𝐽 ∈ Top β†’ ((βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦})βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
108, 9syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ((βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦})βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
116eleq2d 2820 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ βˆͺ 𝐽))
1211biimpa 478 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ βˆͺ 𝐽)
1312snssd 4813 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ {𝑦} βŠ† βˆͺ 𝐽)
142iscld2 22532 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦} βŠ† βˆͺ 𝐽) β†’ ({𝑦} ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽))
158, 13, 14syl2anc 585 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ({𝑦} ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽))
166adantr 482 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
1716eleq2d 2820 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
1817imbi1d 342 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 β†’ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))) ↔ (π‘₯ ∈ βˆͺ 𝐽 β†’ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))))
19 con1b 359 . . . . . . . . 9 ((Β¬ π‘₯ = 𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))) ↔ (Β¬ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})) β†’ π‘₯ = 𝑦))
20 df-ne 2942 . . . . . . . . . 10 (π‘₯ β‰  𝑦 ↔ Β¬ π‘₯ = 𝑦)
2120imbi1i 350 . . . . . . . . 9 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))) ↔ (Β¬ π‘₯ = 𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
22 disjsn 4716 . . . . . . . . . . . . . . 15 ((π‘œ ∩ {𝑦}) = βˆ… ↔ Β¬ 𝑦 ∈ π‘œ)
23 elssuni 4942 . . . . . . . . . . . . . . . 16 (π‘œ ∈ 𝐽 β†’ π‘œ βŠ† βˆͺ 𝐽)
24 reldisj 4452 . . . . . . . . . . . . . . . 16 (π‘œ βŠ† βˆͺ 𝐽 β†’ ((π‘œ ∩ {𝑦}) = βˆ… ↔ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (π‘œ ∈ 𝐽 β†’ ((π‘œ ∩ {𝑦}) = βˆ… ↔ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
2622, 25bitr3id 285 . . . . . . . . . . . . . 14 (π‘œ ∈ 𝐽 β†’ (Β¬ 𝑦 ∈ π‘œ ↔ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
2726anbi2d 630 . . . . . . . . . . . . 13 (π‘œ ∈ 𝐽 β†’ ((π‘₯ ∈ π‘œ ∧ Β¬ 𝑦 ∈ π‘œ) ↔ (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
2827rexbiia 3093 . . . . . . . . . . . 12 (βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ Β¬ 𝑦 ∈ π‘œ) ↔ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
29 rexanali 3103 . . . . . . . . . . . 12 (βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ Β¬ 𝑦 ∈ π‘œ) ↔ Β¬ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ))
3028, 29bitr3i 277 . . . . . . . . . . 11 (βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})) ↔ Β¬ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ))
3130con2bii 358 . . . . . . . . . 10 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) ↔ Β¬ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
3231imbi1i 350 . . . . . . . . 9 ((βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ (Β¬ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})) β†’ π‘₯ = 𝑦))
3319, 21, 323bitr4ri 304 . . . . . . . 8 ((βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
3433imbi2i 336 . . . . . . 7 ((π‘₯ ∈ 𝑋 β†’ (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)) ↔ (π‘₯ ∈ 𝑋 β†’ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))))
35 eldifsn 4791 . . . . . . . . 9 (π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦}) ↔ (π‘₯ ∈ βˆͺ 𝐽 ∧ π‘₯ β‰  𝑦))
3635imbi1i 350 . . . . . . . 8 ((π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦}) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))) ↔ ((π‘₯ ∈ βˆͺ 𝐽 ∧ π‘₯ β‰  𝑦) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
37 impexp 452 . . . . . . . 8 (((π‘₯ ∈ βˆͺ 𝐽 ∧ π‘₯ β‰  𝑦) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))) ↔ (π‘₯ ∈ βˆͺ 𝐽 β†’ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))))
3836, 37bitri 275 . . . . . . 7 ((π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦}) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))) ↔ (π‘₯ ∈ βˆͺ 𝐽 β†’ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))))
3918, 34, 383bitr4g 314 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 β†’ (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)) ↔ (π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦}) β†’ βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))))
4039ralbidv2 3174 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦})βˆƒπ‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
4110, 15, 403bitr4d 311 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ({𝑦} ∈ (Clsdβ€˜π½) ↔ βˆ€π‘₯ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
4241ralbidva 3176 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘¦ ∈ 𝑋 {𝑦} ∈ (Clsdβ€˜π½) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
43 ralcom 3287 . . 3 (βˆ€π‘¦ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
4442, 43bitrdi 287 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘¦ ∈ 𝑋 {𝑦} ∈ (Clsdβ€˜π½) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
455, 7, 443bitr2d 307 1 (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)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412  Clsdccld 22520  Frect1 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17389  df-top 22396  df-topon 22413  df-cld 22523  df-t1 22818
This theorem is referenced by:  t1t0  22852  ist1-3  22853  haust1  22856  t1sep2  22873  isr0  23241  tgpt0  23623
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