Theorem pconntop 34682

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 2731	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispconn 34680	. 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 2105  ∀wral 3060  ∃wrex 3069  ∪ cuni 4908  ‘cfv 6543  (class class class)co 7412  0cc0 11116  1c1 11117  Topctop 22716   Cn ccn 23049  IIcii 24716  PConncpconn 34676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-pconn 34678

This theorem is referenced by:  sconntop  34685  pconnconn  34688  txpconn  34689  ptpconn  34690  qtoppconn  34693  pconnpi1  34694  sconnpi1  34696  cvxsconn  34700