Theorem sconnpconn 35423

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 35422	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 497	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {csn 4581   class class class wbr 5099   × cxp 5623  ‘cfv 6493  (class class class)co 7360  0cc0 11030  1c1 11031  [,]cicc 13268   Cn ccn 23172  IIcii 24828   ≃_phcphtpc 24928  PConncpconn 35415  SConncsconn 35416

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 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-sconn 35418

This theorem is referenced by:  sconntop  35424  txsconn  35437  resconn  35442  iinllyconn  35450  cvmlift2lem10  35508  cvmlift3lem2  35516  cvmlift3  35524