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Theorem pconntop 34514
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 2730 . . 3 𝐽 = 𝐽
21ispconn 34512 . 2 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simplbi 496 1 (𝐽 ∈ PConn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  wrex 3068   cuni 4907  cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113  Topctop 22615   Cn ccn 22948  IIcii 24615  PConncpconn 34508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-pconn 34510
This theorem is referenced by:  sconntop  34517  pconnconn  34520  txpconn  34521  ptpconn  34522  qtoppconn  34525  pconnpi1  34526  sconnpi1  34528  cvxsconn  34532
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