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Theorem sconpcon 28936
Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconpcon  |-  ( J  e. SCon  ->  J  e. PCon )

Proof of Theorem sconpcon
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 isscon 28935 . 2  |-  ( J  e. SCon 
<->  ( J  e. PCon  /\  A. f  e.  ( II 
Cn  J ) ( ( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) ) )
21simplbi 458 1  |-  ( J  e. SCon  ->  J  e. PCon )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = 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 19892   IIcii 21545    ~=ph cphtpc 21635  PConcpcon 28928  SConcscon 28929
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 28931
This theorem is referenced by:  scontop  28937  txscon  28950  rescon  28955  iinllycon  28963  cvmlift2lem10  29021  cvmlift3lem2  29029  cvmlift3  29037
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