MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  concompss Unicode version

Theorem concompss 17175
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 955 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  X
)
2 contop 17159 . . . . . . 7  |-  ( ( Jt  T )  e.  Con  ->  ( Jt  T )  e.  Top )
323ad2ant3 978 . . . . . 6  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( Jt  T )  e.  Top )
4 restrcl 16904 . . . . . . 7  |-  ( ( Jt  T )  e.  Top  ->  ( J  e.  _V  /\  T  e.  _V )
)
54simprd 449 . . . . . 6  |-  ( ( Jt  T )  e.  Top  ->  T  e.  _V )
6 elpwg 3645 . . . . . 6  |-  ( T  e.  _V  ->  ( T  e.  ~P X  <->  T 
C_  X ) )
73, 5, 63syl 18 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( T  e. 
~P X  <->  T  C_  X
) )
81, 7mpbird 223 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  ~P X )
9 3simpc 954 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( A  e.  T  /\  ( Jt  T )  e.  Con )
)
10 eleq2 2357 . . . . . 6  |-  ( y  =  T  ->  ( A  e.  y  <->  A  e.  T ) )
11 oveq2 5882 . . . . . . 7  |-  ( y  =  T  ->  ( Jt  y )  =  ( Jt  T ) )
1211eleq1d 2362 . . . . . 6  |-  ( y  =  T  ->  (
( Jt  y )  e. 
Con 
<->  ( Jt  T )  e.  Con ) )
1310, 12anbi12d 691 . . . . 5  |-  ( y  =  T  ->  (
( A  e.  y  /\  ( Jt  y )  e.  Con )  <->  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
14 eleq2 2357 . . . . . . 7  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
15 oveq2 5882 . . . . . . . 8  |-  ( x  =  y  ->  ( Jt  x )  =  ( Jt  y ) )
1615eleq1d 2362 . . . . . . 7  |-  ( x  =  y  ->  (
( Jt  x )  e.  Con  <->  ( Jt  y )  e.  Con ) )
1714, 16anbi12d 691 . . . . . 6  |-  ( x  =  y  ->  (
( A  e.  x  /\  ( Jt  x )  e.  Con ) 
<->  ( A  e.  y  /\  ( Jt  y )  e.  Con ) ) )
1817cbvrabv 2800 . . . . 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 2938 . . . 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 645 . . 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 3871 . . 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 15 . 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 3238 1  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843  (class class class)co 5874   ↾t crest 13341   Topctop 16647   Conccon 17153
This theorem is referenced by:  concompcld  17176  tgpconcompeqg  17810  tgpconcomp  17811
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-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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-rest 13343  df-top 16652  df-con 17154
  Copyright terms: Public domain W3C validator