Theorem sconnpconn 35727

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 35726	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 501	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {csn 4588   class class class wbr 5108   × cxp 5658  ‘cfv 6536  (class class class)co 7412  0cc0 11106  1c1 11107  [,]cicc 13381   Cn ccn 23392  IIcii 25045   ≃_phcphtpc 25139  PConncpconn 35719  SConncsconn 35720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-sconn 35722

This theorem is used by:  sconntop  35728  txsconn  35741  resconn  35746  iinllyconn  35754  cvmlift2lem10  35812  cvmlift3lem2  35820  cvmlift3  35828