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Theorem conncompcld 21609
 Description: The connected component containing 𝐴 is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompcld ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem conncompcld
StepHypRef Expression
1 topontop 21089 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 474 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
3 conncomp.2 . . . . . . 7 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
4 ssrab2 3913 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
5 sspwuni 4833 . . . . . . . 8 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
64, 5mpbi 222 . . . . . . 7 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
73, 6eqsstri 3861 . . . . . 6 𝑆𝑋
8 toponuni 21090 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98adantr 474 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
107, 9syl5sseq 3879 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 𝐽)
11 eqid 2826 . . . . . 6 𝐽 = 𝐽
1211clsss3 21235 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
132, 10, 12syl2anc 581 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1413, 9sseqtr4d 3868 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
1511sscls 21232 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
162, 10, 15syl2anc 581 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
173conncompid 21606 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
1816, 17sseldd 3829 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
19 simpl 476 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
207a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆𝑋)
213conncompconn 21607 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
22 clsconn 21605 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
2319, 20, 21, 22syl3anc 1496 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
243conncompss 21608 . . 3 ((((cls‘𝐽)‘𝑆) ⊆ 𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2514, 18, 23, 24syl3anc 1496 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2611iscld4 21241 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
272, 10, 26syl2anc 581 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
2825, 27mpbird 249 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3122   ⊆ wss 3799  𝒫 cpw 4379  ∪ cuni 4659  ‘cfv 6124  (class class class)co 6906   ↾t crest 16435  Topctop 21069  TopOnctopon 21086  Clsdccld 21192  clsccl 21194  Conncconn 21586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-iin 4744  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-oadd 7831  df-er 8010  df-en 8224  df-fin 8227  df-fi 8587  df-rest 16437  df-topgen 16458  df-top 21070  df-topon 21087  df-bases 21122  df-cld 21195  df-ntr 21196  df-cls 21197  df-conn 21587 This theorem is referenced by:  conncompclo  21610
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