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Theorem sconnpconn 32482
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 32481 . 2 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simplbi 500 1 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wral 3125  {csn 4543   class class class wbr 5042   × cxp 5529  cfv 6331  (class class class)co 7133  0cc0 10515  1c1 10516  [,]cicc 12720   Cn ccn 21808  IIcii 23459  phcphtpc 23553  PConncpconn 32474  SConncsconn 32475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-iota 6290  df-fv 6339  df-ov 7136  df-sconn 32477
This theorem is referenced by:  sconntop  32483  txsconn  32496  resconn  32501  iinllyconn  32509  cvmlift2lem10  32567  cvmlift3lem2  32575  cvmlift3  32583
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