Theorem sconnpconn 31209

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 31208	. 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2	1	simplbi 476	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {csn 4177   class class class wbr 4653   × cxp 5112  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937  [,]cicc 12178   Cn ccn 21028  IIcii 22678   ≃_phcphtpc 22768  PConncpconn 31201  SConncsconn 31202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-sconn 31204

This theorem is referenced by:  sconntop  31210  txsconn  31223  resconn  31228  iinllyconn  31236  cvmlift2lem10  31294  cvmlift3lem2  31302  cvmlift3  31310