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Theorem sconnpconn 34773
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 34772 . 2 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simplbi 497 1 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wral 3056  {csn 4624   class class class wbr 5142   × cxp 5670  cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131  [,]cicc 13351   Cn ccn 23115  IIcii 24782  phcphtpc 24882  PConncpconn 34765  SConncsconn 34766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-sconn 34768
This theorem is referenced by:  sconntop  34774  txsconn  34787  resconn  34792  iinllyconn  34800  cvmlift2lem10  34858  cvmlift3lem2  34866  cvmlift3  34874
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