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Theorem isscon 23757
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 5866 . . 3  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 fveq2 5525 . . . . 5  |-  ( j  =  J  ->  (  ~=ph  `  j )  =  ( 
~=ph  `  J ) )
32breqd 4034 . . . 4  |-  ( j  =  J  ->  (
f (  ~=ph  `  j
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  <-> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
43imbi2d 307 . . 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 2748 . 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 23753 . 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 2925 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   [,]cicc 10659    Cn ccn 16954   IIcii 18379    ~=ph cphtpc 18467  PConcpcon 23750  SConcscon 23751
This theorem is referenced by:  sconpcon  23758  sconpht  23760  sconpi1  23770  txscon  23772  cvxscon  23774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-scon 23753
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