Theorem sconnpconn 35793

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.

Step	Hyp	Ref	Expression
1		issconn 35792	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 502	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {csn 4587   class class class wbr 5107   × cxp 5657  ‘cfv 6537  (class class class)co 7416  0cc0 11127  1c1 11128  [,]cicc 13403   Cn ccn 23450  IIcii 25104   ≃_phcphtpc 25198  PConncpconn 35785  SConncsconn 35786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-sconn 35788

This theorem is used by:  sconntop  35794  txsconn  35807  resconn  35812  iinllyconn  35820  cvmlift2lem10  35878  cvmlift3lem2  35886  cvmlift3  35894