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Theorem sconpcon 27138
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 27137 . 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 460 1  |-  ( J  e. SCon  ->  J  e. PCon )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2736   {csn 3898   class class class wbr 4313    X. cxp 4859   ` cfv 5439  (class class class)co 6112   0cc0 9303   1c1 9304   [,]cicc 11324    Cn ccn 18850   IIcii 20473    ~=ph cphtpc 20563  PConcpcon 27130  SConcscon 27131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115  df-scon 27133
This theorem is referenced by:  scontop  27139  txscon  27152  rescon  27157  iinllycon  27165  cvmlift2lem10  27223  cvmlift3lem2  27231  cvmlift3  27239
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