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Theorem ist1-2 22850
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 22414 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 eqid 2732 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
32ist1 22824 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
43baib 536 . . 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 22415 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76raleqdv 3325 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘¦ ∈ 𝑋 {𝑦} ∈ (Clsdβ€˜π½) ↔ βˆ€π‘¦ ∈ βˆͺ 𝐽{𝑦} ∈ (Clsdβ€˜π½)))
81adantr 481 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝐽 ∈ Top)
9 eltop2 22477 . . . . . 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 2819 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ βˆͺ 𝐽))
1211biimpa 477 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 ∈ βˆͺ 𝐽)
1312snssd 4812 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ {𝑦} βŠ† βˆͺ 𝐽)
142iscld2 22531 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦} βŠ† βˆͺ 𝐽) β†’ ({𝑦} ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽))
158, 13, 14syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ ({𝑦} ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– {𝑦}) ∈ 𝐽))
166adantr 481 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
1716eleq2d 2819 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
1817imbi1d 341 . . . . . . 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 358 . . . . . . . . 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 2941 . . . . . . . . . 10 (π‘₯ β‰  𝑦 ↔ Β¬ π‘₯ = 𝑦)
2120imbi1i 349 . . . . . . . . 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 4715 . . . . . . . . . . . . . . 15 ((π‘œ ∩ {𝑦}) = βˆ… ↔ Β¬ 𝑦 ∈ π‘œ)
23 elssuni 4941 . . . . . . . . . . . . . . . 16 (π‘œ ∈ 𝐽 β†’ π‘œ βŠ† βˆͺ 𝐽)
24 reldisj 4451 . . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . 14 (π‘œ ∈ 𝐽 β†’ (Β¬ 𝑦 ∈ π‘œ ↔ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦})))
2726anbi2d 629 . . . . . . . . . . . . 13 (π‘œ ∈ 𝐽 β†’ ((π‘₯ ∈ π‘œ ∧ Β¬ 𝑦 ∈ π‘œ) ↔ (π‘₯ ∈ π‘œ ∧ π‘œ βŠ† (βˆͺ 𝐽 βˆ– {𝑦}))))
2827rexbiia 3092 . . . . . . . . . . . 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 3102 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 357 . . . . . . . . . 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 349 . . . . . . . . 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 303 . . . . . . . 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 335 . . . . . . 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 4790 . . . . . . . . 9 (π‘₯ ∈ (βˆͺ 𝐽 βˆ– {𝑦}) ↔ (π‘₯ ∈ βˆͺ 𝐽 ∧ π‘₯ β‰  𝑦))
3635imbi1i 349 . . . . . . . 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 451 . . . . . . . 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 274 . . . . . . 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 313 . . . . . 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 3173 . . . . 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 310 . . . 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 3175 . . 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 3286 . . 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 286 . 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 306 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22394  TopOnctopon 22411  Clsdccld 22519  Frect1 22810
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17388  df-top 22395  df-topon 22412  df-cld 22522  df-t1 22817
This theorem is referenced by:  t1t0  22851  ist1-3  22852  haust1  22855  t1sep2  22872  isr0  23240  tgpt0  23622
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