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Theorem concompss 20100
Description: The connected component containing  A is a superset of any other connected set containing  A. (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
concompss  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S
)
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hints:    S( x)    T( x)

Proof of Theorem concompss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 994 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  X
)
2 contop 20084 . . . . . . 7  |-  ( ( Jt  T )  e.  Con  ->  ( Jt  T )  e.  Top )
323ad2ant3 1017 . . . . . 6  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( Jt  T )  e.  Top )
4 restrcl 19825 . . . . . . 7  |-  ( ( Jt  T )  e.  Top  ->  ( J  e.  _V  /\  T  e.  _V )
)
54simprd 461 . . . . . 6  |-  ( ( Jt  T )  e.  Top  ->  T  e.  _V )
6 elpwg 4007 . . . . . 6  |-  ( T  e.  _V  ->  ( T  e.  ~P X  <->  T 
C_  X ) )
73, 5, 63syl 20 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( T  e. 
~P X  <->  T  C_  X
) )
81, 7mpbird 232 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  ~P X )
9 3simpc 993 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( A  e.  T  /\  ( Jt  T )  e.  Con )
)
10 eleq2 2527 . . . . . 6  |-  ( y  =  T  ->  ( A  e.  y  <->  A  e.  T ) )
11 oveq2 6278 . . . . . . 7  |-  ( y  =  T  ->  ( Jt  y )  =  ( Jt  T ) )
1211eleq1d 2523 . . . . . 6  |-  ( y  =  T  ->  (
( Jt  y )  e. 
Con 
<->  ( Jt  T )  e.  Con ) )
1310, 12anbi12d 708 . . . . 5  |-  ( y  =  T  ->  (
( A  e.  y  /\  ( Jt  y )  e.  Con )  <->  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
14 eleq2 2527 . . . . . . 7  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
15 oveq2 6278 . . . . . . . 8  |-  ( x  =  y  ->  ( Jt  x )  =  ( Jt  y ) )
1615eleq1d 2523 . . . . . . 7  |-  ( x  =  y  ->  (
( Jt  x )  e.  Con  <->  ( Jt  y )  e.  Con ) )
1714, 16anbi12d 708 . . . . . 6  |-  ( x  =  y  ->  (
( A  e.  x  /\  ( Jt  x )  e.  Con ) 
<->  ( A  e.  y  /\  ( Jt  y )  e.  Con ) ) )
1817cbvrabv 3105 . . . . 5  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  =  {
y  e.  ~P X  |  ( A  e.  y  /\  ( Jt  y )  e.  Con ) }
1913, 18elrab2 3256 . . . 4  |-  ( T  e.  { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  <->  ( T  e. 
~P X  /\  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
208, 9, 19sylanbrc 662 . . 3  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
21 elssuni 4264 . . 3  |-  ( T  e.  { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  ->  T  C_  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
2220, 21syl 16 . 2  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  U. {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
23 concomp.2 . 2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
2422, 23syl6sseqr 3536 1  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235  (class class class)co 6270   ↾t crest 14910   Topctop 19561   Conccon 20078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-rest 14912  df-top 19566  df-con 20079
This theorem is referenced by:  concompcld  20101  tgpconcompeqg  20776  tgpconcomp  20777
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