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Theorem hauscmp 23431
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 1136 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆𝑋)
2 hauscmp.1 . . . . . 6 𝑋 = 𝐽
3 eqid 2735 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4 simpl1 1190 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝐽 ∈ Haus)
5 simpl2 1191 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑆𝑋)
6 simpl3 1192 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (𝐽t 𝑆) ∈ Comp)
7 simpr 484 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑥 ∈ (𝑋𝑆))
82, 3, 4, 5, 6, 7hauscmplem 23430 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
9 haustop 23355 . . . . . . . . . . 11 (𝐽 ∈ Haus → 𝐽 ∈ Top)
1093ad2ant1 1132 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11 elssuni 4942 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
1211, 2sseqtrrdi 4047 . . . . . . . . . 10 (𝑧𝐽𝑧𝑋)
132sscls 23080 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
1410, 12, 13syl2an 596 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15 sstr2 4002 . . . . . . . . 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 612 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1817reximdva 3166 . . . . . 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 3144 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
22 eltop2 22998 . . . 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 257 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (𝑋𝑆) ∈ 𝐽)
252iscld 23051 . . 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 713 1 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cdif 3960  wss 3963   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  Clsdccld 23040  clsccl 23042  Hauscha 23332  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cls 23045  df-haus 23339  df-cmp 23411
This theorem is referenced by:  txkgen  23676  cmphaushmeo  23824  cnheibor  25001
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