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Theorem hauscmp 22001
Description: A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)
Hypothesis
Ref Expression
hauscmp.1 𝑋 = 𝐽
Assertion
Ref Expression
hauscmp ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Proof of Theorem hauscmp
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1134 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆𝑋)
2 hauscmp.1 . . . . . 6 𝑋 = 𝐽
3 eqid 2824 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4 simpl1 1188 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝐽 ∈ Haus)
5 simpl2 1189 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑆𝑋)
6 simpl3 1190 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (𝐽t 𝑆) ∈ Comp)
7 simpr 488 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑥 ∈ (𝑋𝑆))
82, 3, 4, 5, 6, 7hauscmplem 22000 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
9 haustop 21925 . . . . . . . . . . 11 (𝐽 ∈ Haus → 𝐽 ∈ Top)
1093ad2ant1 1130 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11 elssuni 4849 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
1211, 2sseqtrrdi 4002 . . . . . . . . . 10 (𝑧𝐽𝑧𝑋)
132sscls 21650 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
1410, 12, 13syl2an 598 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15 sstr2 3958 . . . . . . . . 9 (𝑧 ⊆ ((cls‘𝐽)‘𝑧) → (((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆) → 𝑧 ⊆ (𝑋𝑆)))
1614, 15syl 17 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → (((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆) → 𝑧 ⊆ (𝑋𝑆)))
1716anim2d 614 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1817reximdva 3266 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1918adantr 484 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
208, 19mpd 15 . . . 4 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
2120ralrimiva 3176 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
22 eltop2 21569 . . . 4 (𝐽 ∈ Top → ((𝑋𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
2310, 22syl 17 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ((𝑋𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
2421, 23mpbird 260 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (𝑋𝑆) ∈ 𝐽)
252iscld 21621 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
2610, 25syl 17 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
271, 24, 26mpbir2and 712 1 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  wrex 3133  {crab 3136  cdif 3915  wss 3918   cuni 4819  cfv 6336  (class class class)co 7138  t crest 16683  Topctop 21487  Clsdccld 21610  clsccl 21612  Hauscha 21902  Compccmp 21980
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-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  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 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-iin 4903  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-fin 8496  df-fi 8859  df-rest 16685  df-topgen 16706  df-top 21488  df-topon 21505  df-bases 21540  df-cld 21613  df-cls 21615  df-haus 21909  df-cmp 21981
This theorem is referenced by:  txkgen  22246  cmphaushmeo  22394  cnheibor  23549
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