Theorem issconn 32717

Description: The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
issconn	⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))

Distinct variable group:   𝑓,𝐽

Proof of Theorem issconn

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7164	. . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		fveq2 6663	. . . . 5 ⊢ (𝑗 = 𝐽 → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
3	2	breqd 5047	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))
4	3	imbi2d 344	. . 3 ⊢ (𝑗 = 𝐽 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
5	1, 4	raleqbidv 3319	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
6		df-sconn 32713	. 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
7	5, 6	elrab2 3607	1 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {csn 4525   class class class wbr 5036   × cxp 5526  ‘cfv 6340  (class class class)co 7156  0cc0 10588  1c1 10589  [,]cicc 12795   Cn ccn 21938  IIcii 23590   ≃_phcphtpc 23684  PConncpconn 32710  SConncsconn 32711

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-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-sconn 32713

This theorem is referenced by:  sconnpconn  32718  sconnpht  32720  sconnpi1  32730  txsconn  32732  cvxsconn  32734