Theorem ishtpy 24942

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 24940	. . . . . 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 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
4		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
6	3	oveq1d 7434	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
7	6, 4	oveq12d 7437	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
8	3	unieqd 4922	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
9		ishtpy.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
10		toponuni 22860	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑋 = ∪ 𝐽)
13	8, 12	eqtr4d 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
14	13	raleqdv 3314	. . . . . . 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 3436	. . . . . 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 7495	. . . . 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 22859	. . . . . 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 23189	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
22		ovex 7452	. . . . . . . . . 10 ⊢ ((𝐽 ×t II) Cn 𝐾) ∈ V
23		ssrab2 4073	. . . . . . . . . 10 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
24	22, 23	elpwi2 5349	. . . . . . . . 9 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
25	24	rgen2w 3055	. . . . . . . 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 2725	. . . . . . . . 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 8073	. . . . . . . 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 7452	. . . . . . . 8 ⊢ (𝐽 Cn 𝐾) ∈ V
30	29, 29	xpex 7756	. . . . . . 7 ⊢ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
31	22	pwex 5380	. . . . . . 7 ⊢ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32		fex2 7942	. . . . . . 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 1457	. . . . . 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 7573	. . . 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 6895	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑠) = (𝐹‘𝑠))
37	36	eqeq2d 2736	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑠ℎ0) = (𝑓‘𝑠) ↔ (𝑠ℎ0) = (𝐹‘𝑠)))
38		fveq1 6895	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑠) = (𝐺‘𝑠))
39	38	eqeq2d 2736	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑠ℎ1) = (𝑔‘𝑠) ↔ (𝑠ℎ1) = (𝐺‘𝑠)))
40	37, 39	bi2anan9 636	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
41	40	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠)) ↔ ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))))
42	41	ralbidv 3167	. . . . 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 3426	. . . 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 5335	. . . . 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 7573	. . 3 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
48	47	eleq2d 2811	. 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 7425	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ0) = (𝑠𝐻0))
50	49	eqeq1d 2727	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ0) = (𝐹‘𝑠) ↔ (𝑠𝐻0) = (𝐹‘𝑠)))
51		oveq 7425	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ1) = (𝑠𝐻1))
52	51	eqeq1d 2727	. . . . 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 3167	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
55	54	elrab 3679	. 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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {crab 3418  Vcvv 3461  𝒫 cpw 4604  ∪ cuni 4909   × cxp 5676  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  0cc0 11140  1c1 11141  Topctop 22839  TopOnctopon 22856   Cn ccn 23172   ×_t ctx 23508  IIcii 24839   Htpy chtpy 24937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-top 22840  df-topon 22857  df-cn 23175  df-htpy 24940

This theorem is referenced by:  htpycn  24943  htpyi  24944  ishtpyd  24945