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Theorem pconntop 35210
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 2735 . . 3 𝐽 = 𝐽
21ispconn 35208 . 2 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
32simplbi 497 1 (𝐽 ∈ PConn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068   cuni 4912  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  Topctop 22915   Cn ccn 23248  IIcii 24915  PConncpconn 35204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-pconn 35206
This theorem is referenced by:  sconntop  35213  pconnconn  35216  txpconn  35217  ptpconn  35218  qtoppconn  35221  pconnpi1  35222  sconnpi1  35224  cvxsconn  35228
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