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Theorem concompclo 17490
Description: The connected component containing  A is a subset of any clopen set containing  A. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypothesis
Ref Expression
concomp.2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
Assertion
Ref Expression
concompclo  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  T
)
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hints:    S( x)    T( x)

Proof of Theorem concompclo
StepHypRef Expression
1 eqid 2435 . 2  |-  U. J  =  U. J
2 simp1 957 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  J  e.  (TopOn `  X ) )
3 inss1 3553 . . . . . . 7  |-  ( J  i^i  ( Clsd `  J
) )  C_  J
4 simp2 958 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( J  i^i  ( Clsd `  J ) ) )
53, 4sseldi 3338 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  J )
6 toponss 16986 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  J )  ->  T  C_  X )
72, 5, 6syl2anc 643 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  C_  X
)
8 simp3 959 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  T )
97, 8sseldd 3341 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  X )
10 concomp.2 . . . . 5  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
1110concompcld 17489 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
122, 9, 11syl2anc 643 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  e.  ( Clsd `  J )
)
131cldss 17085 . . 3  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1412, 13syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  U. J
)
1510concompcon 17487 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
162, 9, 15syl2anc 643 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( Jt  S
)  e.  Con )
1710concompid 17486 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
182, 9, 17syl2anc 643 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  S )
19 inelcm 3674 . . 3  |-  ( ( A  e.  T  /\  A  e.  S )  ->  ( T  i^i  S
)  =/=  (/) )
208, 18, 19syl2anc 643 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( T  i^i  S )  =/=  (/) )
21 inss2 3554 . . 3  |-  ( J  i^i  ( Clsd `  J
) )  C_  ( Clsd `  J )
2221, 4sseldi 3338 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( Clsd `  J )
)
231, 14, 16, 5, 20, 22consubclo 17479 1  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   ` cfv 5446  (class class class)co 6073   ↾t crest 13640  TopOnctopon 16951   Clsdccld 17072   Conccon 17466
This theorem is referenced by:  tgpconcompss  18135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-ntr 17076  df-cls 17077  df-con 17467
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