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Theorem conncompcld 23358
Description: The connected component containing 𝐴 is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
Assertion
Ref Expression
conncompcld ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 ∈ (Clsdβ€˜π½))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐽   π‘₯,𝑋
Allowed substitution hint:   𝑆(π‘₯)
Proof of Theorem conncompcld
StepHypRef Expression
1 topontop 22835 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 conncomp.2 . . . . . . 7 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
3 ssrab2 4077 . . . . . . . 8 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋
4 sspwuni 5107 . . . . . . . 8 ({π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋 ↔ βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝑋)
53, 4mpbi 229 . . . . . . 7 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝑋
62, 5eqsstri 4016 . . . . . 6 𝑆 βŠ† 𝑋
7 toponuni 22836 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
87adantr 479 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
96, 8sseqtrid 4034 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
10 eqid 2728 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
1110clsss3 22983 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
121, 9, 11syl2an2r 683 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
1312, 8sseqtrrd 4023 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1410sscls 22980 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
151, 9, 14syl2an2r 683 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
162conncompid 23355 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑆)
1715, 16sseldd 3983 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†))
18 simpl 481 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
196a1i 11 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
202conncompconn 23356 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
21 clsconn 23354 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ (𝐽 β†Ύt 𝑆) ∈ Conn) β†’ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Conn)
2218, 19, 20, 21syl3anc 1368 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Conn)
232conncompss 23357 . . 3 ((((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Conn) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆)
2413, 17, 22, 23syl3anc 1368 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆)
2510iscld4 22989 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆))
261, 9, 25syl2an2r 683 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆))
2724, 26mpbird 256 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 ∈ (Clsdβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430   βŠ† wss 3949  π’« cpw 4606  βˆͺ cuni 4912  β€˜cfv 6553  (class class class)co 7426   β†Ύt crest 17409  Topctop 22815  TopOnctopon 22832  Clsdccld 22940  clsccl 22942  Conncconn 23335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-en 8971  df-fin 8974  df-fi 9442  df-rest 17411  df-topgen 17432  df-top 22816  df-topon 22833  df-bases 22869  df-cld 22943  df-ntr 22944  df-cls 22945  df-conn 23336
This theorem is referenced by:  conncompclo  23359
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