Theorem pcontop 30461

Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
pcontop	⊢ (𝐽 ∈ PCon → 𝐽 ∈ Top)

Proof of Theorem pcontop

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispcon 30459	. 2 ⊢ (𝐽 ∈ PCon ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
3	2	simplbi 475	1 ⊢ (𝐽 ∈ PCon → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  Topctop 20517   Cn ccn 20838  IIcii 22486  PConcpcon 30455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-pcon 30457

This theorem is referenced by:  scontop  30464  pconcon  30467  txpcon  30468  ptpcon  30469  qtoppcon  30472  pconpi1  30473  sconpi1  30475  cvxscon  30479