Theorem sconnpconn 33189

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 33188	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 498	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {csn 4561   class class class wbr 5074   × cxp 5587  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872  [,]cicc 13082   Cn ccn 22375  IIcii 24038   ≃_phcphtpc 24132  PConncpconn 33181  SConncsconn 33182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-sconn 33184

This theorem is referenced by:  sconntop  33190  txsconn  33203  resconn  33208  iinllyconn  33216  cvmlift2lem10  33274  cvmlift3lem2  33282  cvmlift3  33290