Theorem issconn 35213

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 7395	. . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		fveq2 6858	. . . . 5 ⊢ (𝑗 = 𝐽 → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
3	2	breqd 5118	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))
4	3	imbi2d 340	. . 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 35209	. 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
7	5, 6	elrab2 3662	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {csn 4589   class class class wbr 5107   × cxp 5636  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069  [,]cicc 13309   Cn ccn 23111  IIcii 24768   ≃_phcphtpc 24868  PConncpconn 35206  SConncsconn 35207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-sconn 35209

This theorem is referenced by:  sconnpconn  35214  sconnpht  35216  sconnpi1  35226  txsconn  35228  cvxsconn  35230