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Theorem conncompcld 22808
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 22285 . . . . 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 4041 . . . . . . . 8 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋
4 sspwuni 5064 . . . . . . . 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 3982 . . . . . 6 𝑆 βŠ† 𝑋
7 toponuni 22286 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
87adantr 482 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
96, 8sseqtrid 4000 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
10 eqid 2733 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
1110clsss3 22433 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
121, 9, 11syl2an2r 684 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
1312, 8sseqtrrd 3989 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1410sscls 22430 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
151, 9, 14syl2an2r 684 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
162conncompid 22805 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑆)
1715, 16sseldd 3949 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†))
18 simpl 484 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
196a1i 11 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
202conncompconn 22806 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
21 clsconn 22804 . . . 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 1372 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Conn)
232conncompss 22807 . . 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 1372 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆)
2510iscld4 22439 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑆))
261, 9, 25syl2an2r 684 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3406   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282  Clsdccld 22390  clsccl 22392  Conncconn 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-conn 22786
This theorem is referenced by:  conncompclo  22809
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