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Theorem concompcld 17499
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 16993 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 453 . . . . 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 3430 . . . . . . . 8  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X
5 sspwuni 4178 . . . . . . . 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 201 . . . . . . 7  |-  U. {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  X
73, 6eqsstri 3380 . . . . . 6  |-  S  C_  X
8 toponuni 16994 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 453 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  X  =  U. J )
107, 9syl5sseq 3398 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_ 
U. J )
11 eqid 2438 . . . . . 6  |-  U. J  =  U. J
1211clsss3 17125 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
132, 10, 12syl2anc 644 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_ 
U. J )
1413, 9sseqtr4d 3387 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  X )
1511sscls 17122 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
162, 10, 15syl2anc 644 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  ( ( cls `  J
) `  S )
)
173concompid 17496 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
1816, 17sseldd 3351 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( ( cls `  J
) `  S )
)
19 simpl 445 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
207a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  X )
213concompcon 17497 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
22 clscon 17495 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  ( Jt  S )  e.  Con )  ->  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )
2319, 20, 21, 22syl3anc 1185 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  ( ( cls `  J
) `  S )
)  e.  Con )
243concompss 17498 . . 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 1185 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  S )
2611iscld4 17131 . . 3  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( S  e.  ( Clsd `  J
)  <->  ( ( cls `  J ) `  S
)  C_  S )
)
272, 10, 26syl2anc 644 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( cls `  J ) `  S )  C_  S
) )
2825, 27mpbird 225 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5456  (class class class)co 6083   ↾t crest 13650   Topctop 16960  TopOnctopon 16961   Clsdccld 17082   clsccl 17084   Conccon 17476
This theorem is referenced by:  concompclo  17500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-oadd 6730  df-er 6907  df-en 7112  df-fin 7115  df-fi 7418  df-rest 13652  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-con 17477
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