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Theorem concompcld 17176
Description: The connected component containing  A is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
concomp.2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
Assertion
Ref Expression
concompcld  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hint:    S( x)

Proof of Theorem concompcld
StepHypRef Expression
1 topontop 16680 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 451 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  Top )
3 concomp.2 . . . . . . 7  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
4 ssrab2 3271 . . . . . . . 8  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X
5 sspwuni 4003 . . . . . . . 8  |-  ( { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X  <->  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  X )
64, 5mpbi 199 . . . . . . 7  |-  U. {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  X
73, 6eqsstri 3221 . . . . . 6  |-  S  C_  X
8 toponuni 16681 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 451 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  X  =  U. J )
107, 9syl5sseq 3239 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_ 
U. J )
11 eqid 2296 . . . . . 6  |-  U. J  =  U. J
1211clsss3 16812 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
132, 10, 12syl2anc 642 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_ 
U. J )
1413, 9sseqtr4d 3228 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  X )
1511sscls 16809 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
162, 10, 15syl2anc 642 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  ( ( cls `  J
) `  S )
)
173concompid 17173 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
1816, 17sseldd 3194 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( ( cls `  J
) `  S )
)
19 simpl 443 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
207a1i 10 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  X )
213concompcon 17174 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
22 clscon 17172 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  ( Jt  S )  e.  Con )  ->  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )
2319, 20, 21, 22syl3anc 1182 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  ( ( cls `  J
) `  S )
)  e.  Con )
243concompss 17175 . . 3  |-  ( ( ( ( cls `  J
) `  S )  C_  X  /\  A  e.  ( ( cls `  J
) `  S )  /\  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )  ->  ( ( cls `  J ) `  S
)  C_  S )
2514, 18, 23, 24syl3anc 1182 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  S )
2611iscld4 16818 . . 3  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( S  e.  ( Clsd `  J
)  <->  ( ( cls `  J ) `  S
)  C_  S )
)
272, 10, 26syl2anc 642 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( cls `  J ) `  S )  C_  S
) )
2825, 27mpbird 223 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Clsdccld 16769   clsccl 16771   Conccon 17153
This theorem is referenced by:  concompclo  17177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-con 17154
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