Theorem sconnpconn 32474

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 32473	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 500	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {csn 4567   class class class wbr 5066   × cxp 5553  ‘cfv 6355  (class class class)co 7156  0cc0 10537  1c1 10538  [,]cicc 12742   Cn ccn 21832  IIcii 23483   ≃_phcphtpc 23573  PConncpconn 32466  SConncsconn 32467

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 2793

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-sconn 32469

This theorem is referenced by:  sconntop  32475  txsconn  32488  resconn  32493  iinllyconn  32501  cvmlift2lem10  32559  cvmlift3lem2  32567  cvmlift3  32575