MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hauscmp Structured version   Visualization version   GIF version

Theorem hauscmp 21588
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 1171 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆𝑋)
2 hauscmp.1 . . . . . 6 𝑋 = 𝐽
3 eqid 2825 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4 simpl1 1246 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝐽 ∈ Haus)
5 simpl2 1248 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑆𝑋)
6 simpl3 1250 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (𝐽t 𝑆) ∈ Comp)
7 simpr 479 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑥 ∈ (𝑋𝑆))
82, 3, 4, 5, 6, 7hauscmplem 21587 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
9 haustop 21513 . . . . . . . . . . 11 (𝐽 ∈ Haus → 𝐽 ∈ Top)
1093ad2ant1 1167 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11 elssuni 4691 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
1211, 2syl6sseqr 3877 . . . . . . . . . 10 (𝑧𝐽𝑧𝑋)
132sscls 21238 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
1410, 12, 13syl2an 589 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15 sstr2 3834 . . . . . . . . 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 605 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1817reximdva 3225 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1918adantr 474 . . . . 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 3175 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
22 eltop2 21157 . . . 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 249 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (𝑋𝑆) ∈ 𝐽)
252iscld 21209 . . 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 704 1 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  {crab 3121  cdif 3795  wss 3798   cuni 4660  cfv 6127  (class class class)co 6910  t crest 16441  Topctop 21075  Clsdccld 21198  clsccl 21200  Hauscha 21490  Compccmp 21567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-fin 8232  df-fi 8592  df-rest 16443  df-topgen 16464  df-top 21076  df-topon 21093  df-bases 21128  df-cld 21201  df-cls 21203  df-haus 21497  df-cmp 21568
This theorem is referenced by:  txkgen  21833  cmphaushmeo  21981  cnheibor  23131
  Copyright terms: Public domain W3C validator