Theorem ishtpy 24896

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 24894	. . . . . 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 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
4		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7364	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
6	3	oveq1d 7361	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
7	6, 4	oveq12d 7364	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
8	3	unieqd 4872	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
9		ishtpy.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
10		toponuni 22827	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑋 = ∪ 𝐽)
13	8, 12	eqtr4d 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
14	13	raleqdv 3292	. . . . . . 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 3413	. . . . . 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	mpoeq123dv 7421	. . . . 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 22826	. . . . . 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 23154	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
22		ovex 7379	. . . . . . . . . 10 ⊢ ((𝐽 ×t II) Cn 𝐾) ∈ V
23		ssrab2 4030	. . . . . . . . . 10 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
24	22, 23	elpwi2 5273	. . . . . . . . 9 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
25	24	rgen2w 3052	. . . . . . . 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)
26		eqid 2731	. . . . . . . . 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))})
27	26	fmpo 8000	. . . . . . . 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))
28	25, 27	mpbi 230	. . . . . . 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)
29		ovex 7379	. . . . . . . 8 ⊢ (𝐽 Cn 𝐾) ∈ V
30	29, 29	xpex 7686	. . . . . . 7 ⊢ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
31	22	pwex 5318	. . . . . . 7 ⊢ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32		fex2 7866	. . . . . . 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)
33	28, 30, 31, 32	mp3an 1463	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾), 𝑔 ∈ (𝐽 Cn 𝐾) ↦ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}) ∈ V
34	33	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)
35	2, 16, 18, 21, 34	ovmpod 7498	. . . 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))}))
36		fveq1 6821	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑠) = (𝐹‘𝑠))
37	36	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑠ℎ0) = (𝑓‘𝑠) ↔ (𝑠ℎ0) = (𝐹‘𝑠)))
38		fveq1 6821	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑠) = (𝐺‘𝑠))
39	38	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑠ℎ1) = (𝑔‘𝑠) ↔ (𝑠ℎ1) = (𝐺‘𝑠)))
40	37, 39	bi2anan9 638	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
41	40	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
42	41	ralbidv 3155	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
43	42	rabbidv 3402	. . . 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))})
44		ishtpy.4	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
45	22	rabex 5277	. . . . 5 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))} ∈ V
46	45	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))} ∈ V)
47	35, 43, 19, 44, 46	ovmpod 7498	. . 3 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
48	47	eleq2d 2817	. 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ 𝐻 ∈ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))}))
49		oveq 7352	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ0) = (𝑠𝐻0))
50	49	eqeq1d 2733	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ0) = (𝐹‘𝑠) ↔ (𝑠𝐻0) = (𝐹‘𝑠)))
51		oveq 7352	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ1) = (𝑠𝐻1))
52	51	eqeq1d 2733	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ1) = (𝐺‘𝑠) ↔ (𝑠𝐻1) = (𝐺‘𝑠)))
53	50, 52	anbi12d 632	. . . 4 ⊢ (ℎ = 𝐻 → (((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
54	53	ralbidv 3155	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
55	54	elrab 3647	. 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))))
56	48, 55	bitrdi 287	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436  𝒫 cpw 4550  ∪ cuni 4859   × cxp 5614  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  0cc0 11003  1c1 11004  Topctop 22806  TopOnctopon 22823   Cn ccn 23137   ×_t ctx 23473  IIcii 24793   Htpy chtpy 24891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-top 22807  df-topon 22824  df-cn 23140  df-htpy 24894

This theorem is referenced by:  htpycn  24897  htpyi  24898  ishtpyd  24899