Theorem sconnpht 35214

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 35211	. . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
2		fveq1 6906	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3		fveq1 6906	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
4	2, 3	eqeq12d 2751	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5		id 22	. . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
6	2	sneqd 4643	. . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
7	6	xpeq2d 5719	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
8	5, 7	breq12d 5161	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))
9	4, 8	imbi12d 344	. . . 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 3619	. . 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 504	. 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 1537   ∈ wcel 2106  ∀wral 3059  {csn 4631   class class class wbr 5148   × cxp 5687  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  [,]cicc 13387   Cn ccn 23248  IIcii 24915   ≃_phcphtpc 25015  PConncpconn 35204  SConncsconn 35205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-iota 6516  df-fv 6571  df-ov 7434  df-sconn 35207

This theorem is referenced by:  sconnpht2  35223  sconnpi1  35224  txsconn  35226