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Theorem isscon 27249
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 6198 . . 3  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 fveq2 5789 . . . . 5  |-  ( j  =  J  ->  (  ~=ph  `  j )  =  ( 
~=ph  `  J ) )
32breqd 4401 . . . 4  |-  ( j  =  J  ->  (
f (  ~=ph  `  j
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
43imbi2d 316 . . 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 3027 . 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 27245 . 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 3216 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 369    = wceq 1370    e. wcel 1758   A.wral 2795   {csn 3975   class class class wbr 4390    X. cxp 4936   ` cfv 5516  (class class class)co 6190   0cc0 9383   1c1 9384   [,]cicc 11404    Cn ccn 18944   IIcii 20567    ~=ph cphtpc 20657  PConcpcon 27242  SConcscon 27243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-ov 6193  df-scon 27245
This theorem is referenced by:  sconpcon  27250  sconpht  27252  sconpi1  27262  txscon  27264  cvxscon  27266
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