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Theorem sconpht 30465
 Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconpht ((𝐽 ∈ SCon ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))

Proof of Theorem sconpht
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 isscon 30462 . . . 4 (𝐽 ∈ SCon ↔ (𝐽 ∈ PCon ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
21simprbi 479 . . 3 (𝐽 ∈ SCon → ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})))
3 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
4 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
53, 4eqeq12d 2625 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
6 id 22 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
73sneqd 4137 . . . . . . 7 (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
87xpeq2d 5063 . . . . . 6 (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
96, 8breq12d 4596 . . . . 5 (𝑓 = 𝐹 → (𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)})))
105, 9imbi12d 333 . . . 4 (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
1110rspccv 3279 . . 3 (∀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)}))))
122, 11syl 17 . 2 (𝐽 ∈ SCon → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
13123imp 1249 1 ((𝐽 ∈ SCon ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  [,]cicc 12049   Cn ccn 20838  IIcii 22486   ≃phcphtpc 22576  PConcpcon 30455  SConcscon 30456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812  df-ov 6552  df-scon 30458 This theorem is referenced by:  sconpht2  30474  sconpi1  30475  txscon  30477
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