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Theorem sconpcon 24914
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 24913 . 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 447 1  |-  ( J  e. SCon  ->  J  e. PCon )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   {csn 3814   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991   [,]cicc 10919    Cn ccn 17288   IIcii 18905    ~=ph cphtpc 18994  PConcpcon 24906  SConcscon 24907
This theorem is referenced by:  scontop  24915  txscon  24928  rescon  24933  iinllycon  24941  cvmlift2lem10  24999  cvmlift3lem2  25007  cvmlift3  25015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-scon 24909
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