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Theorem concompss 19170
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 988 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  X
)
2 contop 19154 . . . . . . 7  |-  ( ( Jt  T )  e.  Con  ->  ( Jt  T )  e.  Top )
323ad2ant3 1011 . . . . . 6  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( Jt  T )  e.  Top )
4 restrcl 18894 . . . . . . 7  |-  ( ( Jt  T )  e.  Top  ->  ( J  e.  _V  /\  T  e.  _V )
)
54simprd 463 . . . . . 6  |-  ( ( Jt  T )  e.  Top  ->  T  e.  _V )
6 elpwg 3977 . . . . . 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 987 . . . 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 6209 . . . . . . 7  |-  ( y  =  T  ->  ( Jt  y )  =  ( Jt  T ) )
1211eleq1d 2523 . . . . . 6  |-  ( y  =  T  ->  (
( Jt  y )  e. 
Con 
<->  ( Jt  T )  e.  Con ) )
1310, 12anbi12d 710 . . . . 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 6209 . . . . . . . 8  |-  ( x  =  y  ->  ( Jt  x )  =  ( Jt  y ) )
1615eleq1d 2523 . . . . . . 7  |-  ( x  =  y  ->  (
( Jt  x )  e.  Con  <->  ( Jt  y )  e.  Con ) )
1714, 16anbi12d 710 . . . . . 6  |-  ( x  =  y  ->  (
( A  e.  x  /\  ( Jt  x )  e.  Con ) 
<->  ( A  e.  y  /\  ( Jt  y )  e.  Con ) ) )
1817cbvrabv 3077 . . . . 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 3226 . . . 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 664 . . 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 4230 . . 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 3512 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    C_ wss 3437   ~Pcpw 3969   U.cuni 4200  (class class class)co 6201   ↾t crest 14479   Topctop 18631   Conccon 19148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-rest 14481  df-top 18636  df-con 19149
This theorem is referenced by:  concompcld  19171  tgpconcompeqg  19815  tgpconcomp  19816
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