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Theorem sconnpht 31514
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 31511 . . 3 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
2 fveq1 6347 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3 fveq1 6347 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
42, 3eqeq12d 2771 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5 id 22 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
62sneqd 4329 . . . . . . 7 (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
76xpeq2d 5292 . . . . . 6 (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
85, 7breq12d 4813 . . . . 5 (𝑓 = 𝐹 → (𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)})))
94, 8imbi12d 333 . . . 4 (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
109rspccv 3442 . . 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 661 . 2 (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
12113imp 1102 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 1072   = wceq 1628  wcel 2135  wral 3046  {csn 4317   class class class wbr 4800   × cxp 5260  cfv 6045  (class class class)co 6809  0cc0 10124  1c1 10125  [,]cicc 12367   Cn ccn 21226  IIcii 22875  phcphtpc 22965  PConncpconn 31504  SConncsconn 31505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-iota 6008  df-fv 6053  df-ov 6812  df-sconn 31507
This theorem is referenced by:  sconnpht2  31523  sconnpi1  31524  txsconn  31526
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