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Theorem sconpht 24916
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 24913 . . . 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 451 . . 3  |-  ( J  e. SCon  ->  A. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
3 fveq1 5727 . . . . . 6  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
4 fveq1 5727 . . . . . 6  |-  ( f  =  F  ->  (
f `  1 )  =  ( F ` 
1 ) )
53, 4eqeq12d 2450 . . . . 5  |-  ( f  =  F  ->  (
( f `  0
)  =  ( f `
 1 )  <->  ( F `  0 )  =  ( F `  1
) ) )
6 id 20 . . . . . 6  |-  ( f  =  F  ->  f  =  F )
73sneqd 3827 . . . . . . 7  |-  ( f  =  F  ->  { ( f `  0 ) }  =  { ( F `  0 ) } )
87xpeq2d 4902 . . . . . 6  |-  ( f  =  F  ->  (
( 0 [,] 1
)  X.  { ( f `  0 ) } )  =  ( ( 0 [,] 1
)  X.  { ( F `  0 ) } ) )
96, 8breq12d 4225 . . . . 5  |-  ( f  =  F  ->  (
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
F (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) ) )
105, 9imbi12d 312 . . . 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 3049 . . 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 16 . 2  |-  ( J  e. SCon  ->  ( F  e.  ( II  Cn  J
)  ->  ( ( F `  0 )  =  ( F ` 
1 )  ->  F
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( F `  0
) } ) ) ) )
13123imp 1147 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 936    = wceq 1652    e. wcel 1725   A.wral 2705   {csn 3814   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991   [,]cicc 10919    Cn ccn 17288   IIcii 18905    ~=ph cphtpc 18994  PConcpcon 24906  SConcscon 24907
This theorem is referenced by:  sconpht2  24925  sconpi1  24926  txscon  24928
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-iota 5418  df-fv 5462  df-ov 6084  df-scon 24909
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