Theorem sconnpconn 35449

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 35448	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 496	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {csn 4582   class class class wbr 5100   × cxp 5632  ‘cfv 6502  (class class class)co 7370  0cc0 11040  1c1 11041  [,]cicc 13278   Cn ccn 23185  IIcii 24841   ≃_phcphtpc 24941  PConncpconn 35441  SConncsconn 35442

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6458  df-fv 6510  df-ov 7373  df-sconn 35444

This theorem is referenced by:  sconntop  35450  txsconn  35463  resconn  35468  iinllyconn  35476  cvmlift2lem10  35534  cvmlift3lem2  35542  cvmlift3  35550