Theorem sconnpht 34970

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( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))

Proof of Theorem sconnpht

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issconn 34967	. . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2		fveq1 6895	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3		fveq1 6895	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
4	2, 3	eqeq12d 2741	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5		id 22	. . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
6	2	sneqd 4642	. . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
7	6	xpeq2d 5708	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
8	5, 7	breq12d 5162	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))
9	4, 8	imbi12d 343	. . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))))
10	9	rspccv 3603	. . 3 ⊢ (∀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)}))))
11	1, 10	simplbiim 503	. 2 ⊢ (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))))
12	11	3imp 1108	1 ⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {csn 4630   class class class wbr 5149   × cxp 5676  ‘cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141  [,]cicc 13362   Cn ccn 23172  IIcii 24839   ≃_phcphtpc 24939  PConncpconn 34960  SConncsconn 34961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-iota 6501  df-fv 6557  df-ov 7422  df-sconn 34963

This theorem is referenced by:  sconnpht2  34979  sconnpi1  34980  txsconn  34982