Theorem ishtpy 22771

Description: Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Hypotheses

Ref	Expression
ishtpy.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ishtpy.3	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))

Assertion

Ref	Expression
ishtpy	⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))))

Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠   𝑋,𝑠

Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem ishtpy

Dummy variables 𝑓 𝑔 ℎ 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-htpy 22769	. . . . . 6 ⊢ Htpy = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ {ℎ ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 ∈ ∪ 𝑗((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}))
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → Htpy = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ {ℎ ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 ∈ ∪ 𝑗((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))})))
3		simprl 794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
4		simprr 796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
6	3	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
7	6, 4	oveq12d 6668	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
8	3	unieqd 4446	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
9		ishtpy.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
10		toponuni 20719	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑋 = ∪ 𝐽)
13	8, 12	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
14	13	raleqdv 3144	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (∀𝑠 ∈ ∪ 𝑗((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))))
15	7, 14	rabeqbidv 3195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → {ℎ ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 ∈ ∪ 𝑗((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))})
16	5, 5, 15	mpt2eq123dv 6717	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑓 ∈ (𝑗 Cn 𝑘), 𝑔 ∈ (𝑗 Cn 𝑘) ↦ {ℎ ∈ ((𝑗 ×t II) Cn 𝑘) ∣ ∀𝑠 ∈ ∪ 𝑗((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}))
17		topontop 20718	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
18	9, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
19		ishtpy.3	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
20		cntop2 21045	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
22		ssrab2 3687	. . . . . . . . . 10 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
23		ovex 6678	. . . . . . . . . . 11 ⊢ ((𝐽 ×t II) Cn 𝐾) ∈ V
24	23	elpw2 4828	. . . . . . . . . 10 ⊢ ({ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾) ↔ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾))
25	22, 24	mpbir 221	. . . . . . . . 9 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
26	25	rgen2w 2925	. . . . . . . 8 ⊢ ∀𝑓 ∈ (𝐽 Cn 𝐾)∀𝑔 ∈ (𝐽 Cn 𝐾){ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
27		eqid 2622	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))})
28	27	fmpt2 7237	. . . . . . . 8 ⊢ (∀𝑓 ∈ (𝐽 Cn 𝐾)∀𝑔 ∈ (𝐽 Cn 𝐾){ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾) ↔ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾))
29	26, 28	mpbi 220	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾)
30		ovex 6678	. . . . . . . 8 ⊢ (𝐽 Cn 𝐾) ∈ V
31	30, 30	xpex 6962	. . . . . . 7 ⊢ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
32	23	pwex 4848	. . . . . . 7 ⊢ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
33		fex2 7121	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}):((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾))⟶𝒫 ((𝐽 ×t II) Cn 𝐾) ∧ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V ∧ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V) → (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) ∈ V)
34	29, 31, 32, 33	mp3an 1424	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) ∈ V
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) ∈ V)
36	2, 16, 18, 21, 35	ovmpt2d 6788	. . . 4 ⊢ (𝜑 → (𝐽 Htpy 𝐾) = (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}))
37		fveq1 6190	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑠) = (𝐹‘𝑠))
38	37	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑠ℎ0) = (𝑓‘𝑠) ↔ (𝑠ℎ0) = (𝐹‘𝑠)))
39		fveq1 6190	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑠) = (𝐺‘𝑠))
40	39	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑠ℎ1) = (𝑔‘𝑠) ↔ (𝑠ℎ1) = (𝐺‘𝑠)))
41	38, 40	bi2anan9 917	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
42	41	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
43	42	ralbidv 2986	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
44	43	rabbidv 3189	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
45		ishtpy.4	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
46	23	rabex 4813	. . . . 5 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))} ∈ V
47	46	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))} ∈ V)
48	36, 44, 19, 45, 47	ovmpt2d 6788	. . 3 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
49	48	eleq2d 2687	. 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))}))
50		oveq 6656	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ0) = (𝑠𝐻0))
51	50	eqeq1d 2624	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ0) = (𝐹‘𝑠) ↔ (𝑠𝐻0) = (𝐹‘𝑠)))
52		oveq 6656	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ1) = (𝑠𝐻1))
53	52	eqeq1d 2624	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ1) = (𝐺‘𝑠) ↔ (𝑠𝐻1) = (𝐺‘𝑠)))
54	51, 53	anbi12d 747	. . . 4 ⊢ (ℎ = 𝐻 → (((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
55	54	ralbidv 2986	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
56	55	elrab 3363	. 2 ⊢ (𝐻 ∈ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))} ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
57	49, 56	syl6bb 276	1 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363  IIcii 22678   Htpy chtpy 22766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-htpy 22769

This theorem is referenced by:  htpycn  22772  htpyi  22773  ishtpyd  22774