Theorem ishtpy 24939

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 24937	. . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
4		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7386	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
6	3	oveq1d 7383	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (𝑗 ×t II) = (𝐽 ×t II))
7	6, 4	oveq12d 7386	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((𝑗 ×t II) Cn 𝑘) = ((𝐽 ×t II) Cn 𝐾))
8	3	unieqd 4878	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
9		ishtpy.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
10		toponuni 22870	. . . . . . . . . . 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 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
14	13	raleqdv 3298	. . . . . . 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 3419	. . . . . 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 7443	. . . . 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 22869	. . . . . 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 23197	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
22		ovex 7401	. . . . . . . . . 10 ⊢ ((𝐽 ×t II) Cn 𝐾) ∈ V
23		ssrab2 4034	. . . . . . . . . 10 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ⊆ ((𝐽 ×t II) Cn 𝐾)
24	22, 23	elpwi2 5282	. . . . . . . . 9 ⊢ {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))} ∈ 𝒫 ((𝐽 ×t II) Cn 𝐾)
25	24	rgen2w 3057	. . . . . . . 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 2737	. . . . . . . . 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 8022	. . . . . . . 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 7401	. . . . . . . 8 ⊢ (𝐽 Cn 𝐾) ∈ V
30	29, 29	xpex 7708	. . . . . . 7 ⊢ ((𝐽 Cn 𝐾) × (𝐽 Cn 𝐾)) ∈ V
31	22	pwex 5327	. . . . . . 7 ⊢ 𝒫 ((𝐽 ×t II) Cn 𝐾) ∈ V
32		fex2 7888	. . . . . . 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 1464	. . . . . 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 7520	. . . 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 6841	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑠) = (𝐹‘𝑠))
37	36	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑠ℎ0) = (𝑓‘𝑠) ↔ (𝑠ℎ0) = (𝐹‘𝑠)))
38		fveq1 6841	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑠) = (𝐺‘𝑠))
39	38	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑠ℎ1) = (𝑔‘𝑠) ↔ (𝑠ℎ1) = (𝐺‘𝑠)))
40	37, 39	bi2anan9 639	. . . . . . 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 3161	. . . . 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 3408	. . . 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 5286	. . . . 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 7520	. . 3 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) = {ℎ ∈ ((𝐽 ×t II) Cn 𝐾) ∣ ∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠))})
48	47	eleq2d 2823	. 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 7374	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ0) = (𝑠𝐻0))
50	49	eqeq1d 2739	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ0) = (𝐹‘𝑠) ↔ (𝑠𝐻0) = (𝐹‘𝑠)))
51		oveq 7374	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑠ℎ1) = (𝑠𝐻1))
52	51	eqeq1d 2739	. . . . 5 ⊢ (ℎ = 𝐻 → ((𝑠ℎ1) = (𝐺‘𝑠) ↔ (𝑠𝐻1) = (𝐺‘𝑠)))
53	50, 52	anbi12d 633	. . . 4 ⊢ (ℎ = 𝐻 → (((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
54	53	ralbidv 3161	. . 3 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝑋 ((𝑠ℎ0) = (𝐹‘𝑠) ∧ (𝑠ℎ1) = (𝐺‘𝑠)) ↔ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
55	54	elrab 3648	. 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 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442  𝒫 cpw 4556  ∪ cuni 4865   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  0cc0 11038  1c1 11039  Topctop 22849  TopOnctopon 22866   Cn ccn 23180   ×_t ctx 23516  IIcii 24836   Htpy chtpy 24934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-top 22850  df-topon 22867  df-cn 23183  df-htpy 24937

This theorem is referenced by:  htpycn  24940  htpyi  24941  ishtpyd  24942