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Theorem sconpcon 23169
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 23168 . 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 446 1  |-  ( J  e. SCon  ->  J  e. PCon )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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 23161  SConcscon 23162
This theorem is referenced by:  scontop  23170  txscon  23183  rescon  23188  iinllycon  23196  cvmlift2lem10  23254  cvmlift3lem2  23262  cvmlift3  23270
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 23164
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