Theorem sconnpht 33191

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 33188	. . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2		fveq1 6773	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3		fveq1 6773	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
4	2, 3	eqeq12d 2754	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5		id 22	. . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
6	2	sneqd 4573	. . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
7	6	xpeq2d 5619	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
8	5, 7	breq12d 5087	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))
9	4, 8	imbi12d 345	. . . 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 3558	. . 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 505	. 2 ⊢ (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))))
12	11	3imp 1110	1 ⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {csn 4561   class class class wbr 5074   × cxp 5587  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872  [,]cicc 13082   Cn ccn 22375  IIcii 24038   ≃_phcphtpc 24132  PConncpconn 33181  SConncsconn 33182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-iota 6391  df-fv 6441  df-ov 7278  df-sconn 33184

This theorem is referenced by:  sconnpht2  33200  sconnpi1  33201  txsconn  33203