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Theorem isscon 24915
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 6091 . . 3  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 fveq2 5730 . . . . 5  |-  ( j  =  J  ->  (  ~=ph  `  j )  =  ( 
~=ph  `  J ) )
32breqd 4225 . . . 4  |-  ( j  =  J  ->  (
f (  ~=ph  `  j
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
43imbi2d 309 . . 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 2918 . 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 24911 . 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 3096 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {csn 3816   class class class wbr 4214    X. cxp 4878   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993   [,]cicc 10921    Cn ccn 17290   IIcii 18907    ~=ph cphtpc 18996  PConcpcon 24908  SConcscon 24909
This theorem is referenced by:  sconpcon  24916  sconpht  24918  sconpi1  24928  txscon  24930  cvxscon  24932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-scon 24911
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