Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sconnpconn Structured version   Visualization version   GIF version

Theorem sconnpconn 32371
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 32370 . 2 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simplbi 498 1 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {csn 4557   class class class wbr 5057   × cxp 5546  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  [,]cicc 12729   Cn ccn 21760  IIcii 23410  phcphtpc 23500  PConncpconn 32363  SConncsconn 32364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-sconn 32366
This theorem is referenced by:  sconntop  32372  txsconn  32385  resconn  32390  iinllyconn  32398  cvmlift2lem10  32456  cvmlift3lem2  32464  cvmlift3  32472
  Copyright terms: Public domain W3C validator