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Theorem hauscmp 22781
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 1138 . 2 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆𝑋)
2 hauscmp.1 . . . . . 6 𝑋 = 𝐽
3 eqid 2733 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝑥𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4 simpl1 1192 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝐽 ∈ Haus)
5 simpl2 1193 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑆𝑋)
6 simpl3 1194 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → (𝐽t 𝑆) ∈ Comp)
7 simpr 486 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → 𝑥 ∈ (𝑋𝑆))
82, 3, 4, 5, 6, 7hauscmplem 22780 . . . . 5 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑥 ∈ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
9 haustop 22705 . . . . . . . . . . 11 (𝐽 ∈ Haus → 𝐽 ∈ Top)
1093ad2ant1 1134 . . . . . . . . . 10 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝐽 ∈ Top)
11 elssuni 4902 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
1211, 2sseqtrrdi 3999 . . . . . . . . . 10 (𝑧𝐽𝑧𝑋)
132sscls 22430 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧𝑋) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
1410, 12, 13syl2an 597 . . . . . . . . 9 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
15 sstr2 3955 . . . . . . . . 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 613 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) ∧ 𝑧𝐽) → ((𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1817reximdva 3162 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → (∃𝑧𝐽 (𝑥𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) → ∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆))))
1918adantr 482 . . . . 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 3140 . . 3 ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑥 ∈ (𝑋𝑆)∃𝑧𝐽 (𝑥𝑧𝑧 ⊆ (𝑋𝑆)))
22 eltop2 22348 . . . 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 22401 . . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  cdif 3911  wss 3914   cuni 4869  cfv 6500  (class class class)co 7361  t crest 17310  Topctop 22265  Clsdccld 22390  clsccl 22392  Hauscha 22682  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-cls 22395  df-haus 22689  df-cmp 22761
This theorem is referenced by:  txkgen  23026  cmphaushmeo  23174  cnheibor  24341
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