Theorem ishtpy 24041

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 24039	. . . . . 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 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
4		simprr 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7273	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
6	3	oveq1d 7270	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
7	6, 4	oveq12d 7273	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
8	3	unieqd 4850	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
9		ishtpy.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
10		toponuni 21971	. . . . . . . . . . 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 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
14	13	raleqdv 3339	. . . . . . 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 3410	. . . . . 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 7328	. . . . 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 21970	. . . . . 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 22300	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
22		ovex 7288	. . . . . . . . . 10 ⊢ ((𝐽 ×t II) Cn 𝐾) ∈ V
23		ssrab2 4009	. . . . . . . . . 10 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
24	22, 23	elpwi2 5265	. . . . . . . . 9 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
25	24	rgen2w 3076	. . . . . . . 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 2738	. . . . . . . . 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 7881	. . . . . . . 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 229	. . . . . . 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 7288	. . . . . . . 8 ⊢ (𝐽 Cn 𝐾) ∈ V
30	29, 29	xpex 7581	. . . . . . 7 ⊢ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
31	22	pwex 5298	. . . . . . 7 ⊢ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32		fex2 7754	. . . . . . 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 1459	. . . . . 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 7403	. . . 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 6755	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑠) = (𝐹‘𝑠))
37	36	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑠ℎ0) = (𝑓‘𝑠) ↔ (𝑠ℎ0) = (𝐹‘𝑠)))
38		fveq1 6755	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑠) = (𝐺‘𝑠))
39	38	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑠ℎ1) = (𝑔‘𝑠) ↔ (𝑠ℎ1) = (𝐺‘𝑠)))
40	37, 39	bi2anan9 635	. . . . . . 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 3120	. . . . 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 3404	. . . 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 5251	. . . . 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 7403	. . 3 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
48	47	eleq2d 2824	. 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 7261	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ0) = (𝑠𝐻0))
50	49	eqeq1d 2740	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ0) = (𝐹‘𝑠) ↔ (𝑠𝐻0) = (𝐹‘𝑠)))
51		oveq 7261	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ1) = (𝑠𝐻1))
52	51	eqeq1d 2740	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ1) = (𝐺‘𝑠) ↔ (𝑠𝐻1) = (𝐺‘𝑠)))
53	50, 52	anbi12d 630	. . . 4 ⊢ (ℎ = 𝐻 → (((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
54	53	ralbidv 3120	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
55	54	elrab 3617	. 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 286	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422  𝒫 cpw 4530  ∪ cuni 4836   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  0cc0 10802  1c1 10803  Topctop 21950  TopOnctopon 21967   Cn ccn 22283   ×_t ctx 22619  IIcii 23944   Htpy chtpy 24036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-top 21951  df-topon 21968  df-cn 22286  df-htpy 24039

This theorem is referenced by:  htpycn  24042  htpyi  24043  ishtpyd  24044