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Theorem concompclo 21048
 Description: The connected component containing 𝐴 is a subset of any clopen set containing 𝐴. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypothesis
Ref Expression
concomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
Assertion
Ref Expression
concompclo ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆𝑇)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem concompclo
StepHypRef Expression
1 eqid 2610 . 2 𝐽 = 𝐽
2 simp1 1054 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘𝑋))
3 inss1 3795 . . . . . . 7 (𝐽 ∩ (Clsd‘𝐽)) ⊆ 𝐽
4 simp2 1055 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)))
53, 4sseldi 3566 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇𝐽)
6 toponss 20544 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇𝐽) → 𝑇𝑋)
72, 5, 6syl2anc 691 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇𝑋)
8 simp3 1056 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑇)
97, 8sseldd 3569 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑋)
10 concomp.2 . . . . 5 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
1110concompcld 21047 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))
122, 9, 11syl2anc 691 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆 ∈ (Clsd‘𝐽))
131cldss 20643 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
1412, 13syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆 𝐽)
1510concompcon 21045 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Con)
162, 9, 15syl2anc 691 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → (𝐽t 𝑆) ∈ Con)
1710concompid 21044 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
182, 9, 17syl2anc 691 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑆)
19 inelcm 3984 . . 3 ((𝐴𝑇𝐴𝑆) → (𝑇𝑆) ≠ ∅)
208, 18, 19syl2anc 691 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → (𝑇𝑆) ≠ ∅)
21 inss2 3796 . . 3 (𝐽 ∩ (Clsd‘𝐽)) ⊆ (Clsd‘𝐽)
2221, 4sseldi 3566 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇 ∈ (Clsd‘𝐽))
231, 14, 16, 5, 20, 22consubclo 21037 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  TopOnctopon 20518  Clsdccld 20630  Conccon 21024 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-oadd 7451  df-er 7629  df-en 7842  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-ntr 20634  df-cls 20635  df-con 21025 This theorem is referenced by:  tgpconcompss  21727
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