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Theorem sconnpht 32590
Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconnpht ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))

Proof of Theorem sconnpht
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 issconn 32587 . . 3 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
2 fveq1 6648 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3 fveq1 6648 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
42, 3eqeq12d 2817 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5 id 22 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
62sneqd 4540 . . . . . . 7 (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
76xpeq2d 5553 . . . . . 6 (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
85, 7breq12d 5046 . . . . 5 (𝑓 = 𝐹 → (𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)})))
94, 8imbi12d 348 . . . 4 (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
109rspccv 3571 . . 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)}))))
111, 10simplbiim 508 . 2 (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
12113imp 1108 1 ((𝐽 ∈ SConn ∧ 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 1084   = wceq 1538  wcel 2112  wral 3109  {csn 4528   class class class wbr 5033   × cxp 5521  cfv 6328  (class class class)co 7139  0cc0 10530  1c1 10531  [,]cicc 12733   Cn ccn 21833  IIcii 23484  phcphtpc 23578  PConncpconn 32580  SConncsconn 32581
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-iota 6287  df-fv 6336  df-ov 7142  df-sconn 32583
This theorem is referenced by:  sconnpht2  32599  sconnpi1  32600  txsconn  32602
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