Theorem pconntop 33087

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 2738	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispconn 33085	. 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 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  Topctop 21950   Cn ccn 22283  IIcii 23944  PConncpconn 33081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-pconn 33083

This theorem is referenced by:  sconntop  33090  pconnconn  33093  txpconn  33094  ptpconn  33095  qtoppconn  33098  pconnpi1  33099  sconnpi1  33101  cvxsconn  33105