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

Theorem hauscmp 21020
 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 1055 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆𝑋)
2 hauscmp.1 . . . . . 6 𝑋 = 𝐽
3 eqid 2610 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4 simpl1 1057 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝐽 ∈ Haus)
5 simpl2 1058 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑆𝑋)
6 simpl3 1059 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (𝐽t 𝑆) ∈ Comp)
7 simpr 476 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑥 ∈ (𝑋𝑆))
82, 3, 4, 5, 6, 7hauscmplem 21019 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
9 haustop 20945 . . . . . . . . . . 11 (𝐽 ∈ Haus → 𝐽 ∈ Top)
1093ad2ant1 1075 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11 elssuni 4403 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
1211, 2syl6sseqr 3615 . . . . . . . . . 10 (𝑧𝐽𝑧𝑋)
132sscls 20670 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
1410, 12, 13syl2an 493 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15 sstr2 3575 . . . . . . . . 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 587 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1817reximdva 3000 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1918adantr 480 . . . . 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 2949 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
22 eltop2 20590 . . . 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 246 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (𝑋𝑆) ∈ 𝐽)
252iscld 20641 . . 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 959 1 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Clsdccld 20630  clsccl 20632  Hauscha 20922  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cls 20635  df-haus 20929  df-cmp 21000 This theorem is referenced by:  txkgen  21265  cmphaushmeo  21413  cnheibor  22562
 Copyright terms: Public domain W3C validator