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Theorem sconnpconn 34287
Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconnpconn (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Proof of Theorem sconnpconn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 issconn 34286 . 2 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simplbi 498 1 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3061  {csn 4628   class class class wbr 5148   × cxp 5674  cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113  [,]cicc 13329   Cn ccn 22735  IIcii 24398  phcphtpc 24492  PConncpconn 34279  SConncsconn 34280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-sconn 34282
This theorem is referenced by:  sconntop  34288  txsconn  34301  resconn  34306  iinllyconn  34314  cvmlift2lem10  34372  cvmlift3lem2  34380  cvmlift3  34388
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