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Theorem sconnpconn 32932
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 32931 . 2 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simplbi 501 1 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2112  wral 3064  {csn 4557   class class class wbr 5069   × cxp 5566  cfv 6400  (class class class)co 7234  0cc0 10758  1c1 10759  [,]cicc 12967   Cn ccn 22152  IIcii 23803  phcphtpc 23897  PConncpconn 32924  SConncsconn 32925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-iota 6358  df-fv 6408  df-ov 7237  df-sconn 32927
This theorem is referenced by:  sconntop  32933  txsconn  32946  resconn  32951  iinllyconn  32959  cvmlift2lem10  33017  cvmlift3lem2  33025  cvmlift3  33033
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