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Theorem conncompclo 21361
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
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompclo ((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 conncompclo
StepHypRef Expression
1 eqid 2724 . 2 𝐽 = 𝐽
2 simp1 1128 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘𝑋))
3 inss1 3941 . . . . . . 7 (𝐽 ∩ (Clsd‘𝐽)) ⊆ 𝐽
4 simp2 1129 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)))
53, 4sseldi 3707 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇𝐽)
6 toponss 20854 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇𝐽) → 𝑇𝑋)
72, 5, 6syl2anc 696 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇𝑋)
8 simp3 1130 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑇)
97, 8sseldd 3710 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑋)
10 conncomp.2 . . . . 5 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
1110conncompcld 21360 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))
122, 9, 11syl2anc 696 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆 ∈ (Clsd‘𝐽))
131cldss 20956 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
1412, 13syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆 𝐽)
1510conncompconn 21358 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
162, 9, 15syl2anc 696 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → (𝐽t 𝑆) ∈ Conn)
1710conncompid 21357 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
182, 9, 17syl2anc 696 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝐴𝑆)
19 inelcm 4140 . . 3 ((𝐴𝑇𝐴𝑆) → (𝑇𝑆) ≠ ∅)
208, 18, 19syl2anc 696 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → (𝑇𝑆) ≠ ∅)
21 inss2 3942 . . 3 (𝐽 ∩ (Clsd‘𝐽)) ⊆ (Clsd‘𝐽)
2221, 4sseldi 3707 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑇 ∈ (Clsd‘𝐽))
231, 14, 16, 5, 20, 22connsubclo 21350 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  {crab 3018  cin 3679  wss 3680  c0 4023  𝒫 cpw 4266   cuni 4544  cfv 6001  (class class class)co 6765  t crest 16204  TopOnctopon 20838  Clsdccld 20943  Conncconn 21337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-oadd 7684  df-er 7862  df-en 8073  df-fin 8076  df-fi 8433  df-rest 16206  df-topgen 16227  df-top 20822  df-topon 20839  df-bases 20873  df-cld 20946  df-ntr 20947  df-cls 20948  df-conn 21338
This theorem is referenced by:  tgpconncompss  22039
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