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Theorem sconpht 23171
Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconpht  |-  ( ( J  e. SCon  /\  F  e.  ( II  Cn  J
)  /\  ( F `  0 )  =  ( F `  1
) )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) )

Proof of Theorem sconpht
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 isscon 23168 . . . 4  |-  ( J  e. SCon 
<->  ( J  e. PCon  /\  A. f  e.  ( II 
Cn  J ) ( ( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) ) )
21simprbi 450 . . 3  |-  ( J  e. SCon  ->  A. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
3 fveq1 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
4 fveq1 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  1 )  =  ( F ` 
1 ) )
53, 4eqeq12d 2297 . . . . 5  |-  ( f  =  F  ->  (
( f `  0
)  =  ( f `
 1 )  <->  ( F `  0 )  =  ( F `  1
) ) )
6 id 19 . . . . . 6  |-  ( f  =  F  ->  f  =  F )
73sneqd 3653 . . . . . . 7  |-  ( f  =  F  ->  { ( f `  0 ) }  =  { ( F `  0 ) } )
87xpeq2d 4713 . . . . . 6  |-  ( f  =  F  ->  (
( 0 [,] 1
)  X.  { ( f `  0 ) } )  =  ( ( 0 [,] 1
)  X.  { ( F `  0 ) } ) )
96, 8breq12d 4036 . . . . 5  |-  ( f  =  F  ->  (
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
F (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) ) )
105, 9imbi12d 311 . . . 4  |-  ( f  =  F  ->  (
( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  <->  ( ( F `  0 )  =  ( F ` 
1 )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) ) ) )
1110rspccv 2881 . . 3  |-  ( A. f  e.  ( II  Cn  J ) ( ( f `  0 )  =  ( f ` 
1 )  ->  f
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) )  ->  ( F  e.  ( II  Cn  J
)  ->  ( ( F `  0 )  =  ( F ` 
1 )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) ) ) )
122, 11syl 15 . 2  |-  ( J  e. SCon  ->  ( F  e.  ( II  Cn  J
)  ->  ( ( F `  0 )  =  ( F ` 
1 )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) ) ) )
13123imp 1145 1  |-  ( ( J  e. SCon  /\  F  e.  ( II  Cn  J
)  /\  ( F `  0 )  =  ( F `  1
) )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   [,]cicc 10659    Cn ccn 16954   IIcii 18379    ~=ph cphtpc 18467  PConcpcon 23161  SConcscon 23162
This theorem is referenced by:  sconpht2  23180  sconpi1  23181  txscon  23183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-iota 5219  df-fv 5263  df-ov 5861  df-scon 23164
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