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Theorem conncompcld 23458
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 22935 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 conncomp.2 . . . . . . 7 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
3 ssrab2 4090 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
4 sspwuni 5105 . . . . . . . 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 4030 . . . . . 6 𝑆𝑋
7 toponuni 22936 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
87adantr 480 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
96, 8sseqtrid 4048 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 𝐽)
10 eqid 2735 . . . . . 6 𝐽 = 𝐽
1110clsss3 23083 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
121, 9, 11syl2an2r 685 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1312, 8sseqtrrd 4037 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
1410sscls 23080 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
151, 9, 14syl2an2r 685 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
162conncompid 23455 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
1715, 16sseldd 3996 . . 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 23456 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
21 clsconn 23454 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
2218, 19, 20, 21syl3anc 1370 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn)
232conncompss 23457 . . 3 ((((cls‘𝐽)‘𝑆) ⊆ 𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Conn) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2413, 17, 22, 23syl3anc 1370 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑆)
2510iscld4 23089 . . 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 1537  wcel 2106  {crab 3433  wss 3963  𝒫 cpw 4605   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  clsccl 23042  Conncconn 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-conn 23436
This theorem is referenced by:  conncompclo  23459
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