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Theorem conncompcld 23293
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 22770 . . . . 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 4072 . . . . . . . 8 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋
4 sspwuni 5096 . . . . . . . 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 4011 . . . . . 6 𝑆 βŠ† 𝑋
7 toponuni 22771 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
87adantr 480 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
96, 8sseqtrid 4029 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
10 eqid 2726 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
1110clsss3 22918 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
121, 9, 11syl2an2r 682 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
1312, 8sseqtrrd 4018 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1410sscls 22915 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
151, 9, 14syl2an2r 682 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
162conncompid 23290 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑆)
1715, 16sseldd 3978 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†))
18 simpl 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
196a1i 11 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
202conncompconn 23291 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
21 clsconn 23289 . . . 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 23292 . . 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 22924 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆))
261, 9, 25syl2an2r 682 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆))
2724, 26mpbird 257 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 395   = wceq 1533   ∈ wcel 2098  {crab 3426   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902  β€˜cfv 6537  (class class class)co 7405   β†Ύt crest 17375  Topctop 22750  TopOnctopon 22767  Clsdccld 22875  clsccl 22877  Conncconn 23270
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17377  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cld 22878  df-ntr 22879  df-cls 22880  df-conn 23271
This theorem is referenced by:  conncompclo  23294
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