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Theorem pconntop 31754
Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
pconntop (𝐽 ∈ PConn → 𝐽 ∈ Top)

Proof of Theorem pconntop
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2826 . . 3 𝐽 = 𝐽
21ispconn 31752 . 2 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simplbi 493 1 (𝐽 ∈ PConn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3118  wrex 3119   cuni 4659  cfv 6124  (class class class)co 6906  0cc0 10253  1c1 10254  Topctop 21069   Cn ccn 21400  IIcii 23049  PConncpconn 31748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-iota 6087  df-fv 6132  df-ov 6909  df-pconn 31750
This theorem is referenced by:  sconntop  31757  pconnconn  31760  txpconn  31761  ptpconn  31762  qtoppconn  31765  pconnpi1  31766  sconnpi1  31768  cvxsconn  31772
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