Theorem sconpcon 30463

Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
sconpcon	⊢ (𝐽 ∈ SCon → 𝐽 ∈ PCon)

Proof of Theorem sconpcon

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isscon 30462	. 2 ⊢ (𝐽 ∈ SCon ↔ (𝐽 ∈ PCon ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 475	1 ⊢ (𝐽 ∈ SCon → 𝐽 ∈ PCon)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  [,]cicc 12049   Cn ccn 20838  IIcii 22486   ≃_phcphtpc 22576  PConcpcon 30455  SConcscon 30456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-scon 30458

This theorem is referenced by:  scontop  30464  txscon  30477  rescon  30482  iinllycon  30490  cvmlift2lem10  30548  cvmlift3lem2  30556  cvmlift3  30564