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Theorem isscon 28938
Description: The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
isscon  |-  ( J  e. SCon 
<->  ( J  e. PCon  /\  A. f  e.  ( II 
Cn  J ) ( ( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) ) )
Distinct variable group:    f, J

Proof of Theorem isscon
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 oveq2 6278 . . 3  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 fveq2 5848 . . . . 5  |-  ( j  =  J  ->  (  ~=ph  `  j )  =  ( 
~=ph  `  J ) )
32breqd 4450 . . . 4  |-  ( j  =  J  ->  (
f (  ~=ph  `  j
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
43imbi2d 314 . . 3  |-  ( j  =  J  ->  (
( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  j ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  <->  ( (
f `  0 )  =  ( f ` 
1 )  ->  f
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ) ) )
51, 4raleqbidv 3065 . 2  |-  ( j  =  J  ->  ( A. f  e.  (
II  Cn  j )
( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  j ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  <->  A. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  ( f `  1
)  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) ) )
6 df-scon 28934 . 2  |- SCon  =  {
j  e. PCon  |  A. f  e.  ( II  Cn  j ) ( ( f `  0 )  =  ( f ` 
1 )  ->  f
(  ~=ph  `  j )
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ) }
75, 6elrab2 3256 1  |-  ( J  e. SCon 
<->  ( J  e. PCon  /\  A. 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    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {csn 4016   class class class wbr 4439    X. cxp 4986   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482   [,]cicc 11535    Cn ccn 19895   IIcii 21548    ~=ph cphtpc 21638  PConcpcon 28931  SConcscon 28932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-scon 28934
This theorem is referenced by:  sconpcon  28939  sconpht  28941  sconpi1  28951  txscon  28953  cvxscon  28955
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