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Theorem sconpht 27133
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 27130 . . . 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 464 . . 3  |-  ( J  e. SCon  ->  A. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
3 fveq1 5705 . . . . . 6  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
4 fveq1 5705 . . . . . 6  |-  ( f  =  F  ->  (
f `  1 )  =  ( F ` 
1 ) )
53, 4eqeq12d 2457 . . . . 5  |-  ( f  =  F  ->  (
( f `  0
)  =  ( f `
 1 )  <->  ( F `  0 )  =  ( F `  1
) ) )
6 id 22 . . . . . 6  |-  ( f  =  F  ->  f  =  F )
73sneqd 3904 . . . . . . 7  |-  ( f  =  F  ->  { ( f `  0 ) }  =  { ( F `  0 ) } )
87xpeq2d 4879 . . . . . 6  |-  ( f  =  F  ->  (
( 0 [,] 1
)  X.  { ( f `  0 ) } )  =  ( ( 0 [,] 1
)  X.  { ( F `  0 ) } ) )
96, 8breq12d 4320 . . . . 5  |-  ( f  =  F  ->  (
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
F (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) ) )
105, 9imbi12d 320 . . . 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 3085 . . 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 1181 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   {csn 3892   class class class wbr 4307    X. cxp 4853   ` cfv 5433  (class class class)co 6106   0cc0 9297   1c1 9298   [,]cicc 11318    Cn ccn 18843   IIcii 20466    ~=ph cphtpc 20556  PConcpcon 27123  SConcscon 27124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-xp 4861  df-iota 5396  df-fv 5441  df-ov 6109  df-scon 27126
This theorem is referenced by:  sconpht2  27142  sconpi1  27143  txscon  27145
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