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Theorem conncompcld 23443
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 22920 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 conncomp.2 . . . . . . 7 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
3 ssrab2 4079 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
4 sspwuni 5099 . . . . . . . 8 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
53, 4mpbi 230 . . . . . . 7 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
62, 5eqsstri 4029 . . . . . 6 𝑆𝑋
7 toponuni 22921 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
87adantr 480 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
96, 8sseqtrid 4025 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 𝐽)
10 eqid 2736 . . . . . 6 𝐽 = 𝐽
1110clsss3 23068 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
121, 9, 11syl2an2r 685 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1312, 8sseqtrrd 4020 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
1410sscls 23065 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
151, 9, 14syl2an2r 685 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
162conncompid 23440 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
1715, 16sseldd 3983 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
18 simpl 482 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
196a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆𝑋)
202conncompconn 23441 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
21 clsconn 23439 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
2218, 19, 20, 21syl3anc 1372 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
232conncompss 23442 . . 3 ((((cls‘𝐽)‘𝑆) ⊆ 𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2413, 17, 22, 23syl3anc 1372 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2510iscld4 23074 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
261, 9, 25syl2an2r 685 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
2724, 26mpbird 257 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  wss 3950  𝒫 cpw 4599   cuni 4906  cfv 6560  (class class class)co 7432  t crest 17466  Topctop 22900  TopOnctopon 22917  Clsdccld 23025  clsccl 23027  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cld 23028  df-ntr 23029  df-cls 23030  df-conn 23421
This theorem is referenced by:  conncompclo  23444
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