Theorem pconntop 35212

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.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispconn 35210	. 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069  Topctop 22780   Cn ccn 23111  IIcii 24768  PConncpconn 35206

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-pconn 35208

This theorem is referenced by:  sconntop  35215  pconnconn  35218  txpconn  35219  ptpconn  35220  qtoppconn  35223  pconnpi1  35224  sconnpi1  35226  cvxsconn  35230