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Theorem issconn 31334
 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( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
Distinct variable group:   𝑓,𝐽

Proof of Theorem issconn
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . 3 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2 fveq2 6229 . . . . 5 (𝑗 = 𝐽 → ( ≃ph𝑗) = ( ≃ph𝐽))
32breqd 4696 . . . 4 (𝑗 = 𝐽 → (𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})))
43imbi2d 329 . . 3 (𝑗 = 𝐽 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
51, 4raleqbidv 3182 . 2 (𝑗 = 𝐽 → (∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
6 df-sconn 31330 . 2 SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}
75, 6elrab2 3399 1 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {csn 4210   class class class wbr 4685   × cxp 5141  ‘cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  [,]cicc 12216   Cn ccn 21076  IIcii 22725   ≃phcphtpc 22815  PConncpconn 31327  SConncsconn 31328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-sconn 31330 This theorem is referenced by:  sconnpconn  31335  sconnpht  31337  sconnpi1  31347  txsconn  31349  cvxsconn  31351
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