Theorem pconntop 32703

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 2758	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ispconn 32701	. 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∪ cuni 4798  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576  Topctop 21593   Cn ccn 21924  IIcii 23576  PConncpconn 32697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153  df-pconn 32699

This theorem is referenced by:  sconntop  32706  pconnconn  32709  txpconn  32710  ptpconn  32711  qtoppconn  32714  pconnpi1  32715  sconnpi1  32717  cvxsconn  32721