Theorem ptrescn 23618

Description: Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)

Hypotheses

Ref	Expression
ptrescn.1	⊢ 𝑋 = ∪ 𝐽
ptrescn.2	⊢ 𝐽 = (∏t‘𝐹)
ptrescn.3	⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐵))

Assertion

Ref	Expression
ptrescn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐾   𝑥,𝑉   𝑥,𝑋

Allowed substitution hint:   𝐽(𝑥)

Proof of Theorem ptrescn

Dummy variables 𝑢 𝑘 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 1195	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ⊆ 𝐴)
2		ptrescn.2	. . . . . . . . . 10 ⊢ 𝐽 = (∏t‘𝐹)
3	2	ptuni 23573	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5		ptrescn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2790	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
7	6	eleq2d 2823	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ∈ 𝑋))
8	7	biimpar 477	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
9		resixp 8876	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
10	1, 8, 9	syl2anc 585	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
11		ixpeq2 8854	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
12		fvres 6855	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
13	12	unieqd 4864	. . . . . . 7 ⊢ (𝑘 ∈ 𝐵 → ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘))
14	11, 13	mprg 3058	. . . . . 6 ⊢ X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘)
15		ssexg 5261	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
16	15	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
17	16	3adant2 1132	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
18		fssres 6702	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
19	18	3adant1 1131	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
20		ptrescn.3	. . . . . . . 8 ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐵))
21	20	ptuni 23573	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
22	17, 19, 21	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
23	14, 22	eqtr3id 2786	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
24	23	adantr 480	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
25	10, 24	eleqtrd 2839	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
26	25	fmpttd 7063	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾)
27		fimacnv 6686	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾 → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) = 𝑋)
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) = 𝑋)
29		pttop 23561	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
30	2, 29	eqeltrid 2841	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 ∈ Top)
31	30	3adant3 1133	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
32	5	topopn 22885	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝑋 ∈ 𝐽)
34	28, 33	eqeltrd 2837	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽)
35		elsni 4585	. . . . . . 7 ⊢ (𝑣 ∈ {∪ 𝐾} → 𝑣 = ∪ 𝐾)
36	35	imaeq2d 6021	. . . . . 6 ⊢ (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾))
37	36	eleq1d 2822	. . . . 5 ⊢ (𝑣 ∈ {∪ 𝐾} → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽))
38	34, 37	syl5ibrcom 247	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
39	38	ralrimiv 3129	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣 ∈ {∪ 𝐾} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
40		imaco 6211	. . . . . . . . 9 ⊢ ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘))) “ 𝑢) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))
41		cnvco 5836	. . . . . . . . . . 11 ⊢ ◡((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
42	25	adantlr 716	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
43		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)))
44		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) = (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
45		fveq1 6835	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 ↾ 𝐵) → (𝑧‘𝑘) = ((𝑥 ↾ 𝐵)‘𝑘))
46	42, 43, 44, 45	fmptco 7078	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = (𝑥 ∈ 𝑋 ↦ ((𝑥 ↾ 𝐵)‘𝑘)))
47		fvres 6855	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐵 → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
48	47	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
49	48	mpteq2dv 5180	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ ((𝑥 ↾ 𝐵)‘𝑘)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)))
50	46, 49	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)))
51	50	cnveqd 5826	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ◡((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = ◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)))
52	41, 51	eqtr3id 2786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘))) = ◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)))
53	52	imaeq1d 6020	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘))) “ 𝑢) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢))
54	40, 53	eqtr3id 2786	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢))
55		simpl1 1193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐴 ∈ 𝑉)
56		simpl2 1194	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐹:𝐴⟶Top)
57		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐵 ⊆ 𝐴)
58		simprl 771	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐵)
59	57, 58	sseldd 3923	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐴)
60	5, 2	ptpjcn 23590	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
61	55, 56, 59, 60	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
62		simprr 773	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑢 ∈ (𝐹‘𝑘))
63		cnima 23244	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑢 ∈ (𝐹‘𝑘)) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
64	61, 62, 63	syl2anc 585	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
65	54, 64	eqeltrd 2837	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ 𝐽)
66		imaeq2 6017	. . . . . . . 8 ⊢ (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
67	66	eleq1d 2822	. . . . . . 7 ⊢ (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ 𝐽))
68	65, 67	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
69	68	rexlimdvva 3195	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (∃𝑘 ∈ 𝐵 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
70	69	alrimiv 1929	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣(∃𝑘 ∈ 𝐵 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
71		eqid 2737	. . . . . . 7 ⊢ (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))
72	71	rnmpo 7495	. . . . . 6 ⊢ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) = {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)}
73	72	raleqi 3294	. . . . 5 ⊢ (∀𝑣 ∈ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ ∀𝑣 ∈ {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
74	12	rexeqdv 3297	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → (∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
75		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ 𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
76	75	rexbidv 3162	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (∃𝑢 ∈ (𝐹‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
77	74, 76	sylan9bbr 510	. . . . . . 7 ⊢ ((𝑦 = 𝑣 ∧ 𝑘 ∈ 𝐵) → (∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
78	77	rexbidva 3160	. . . . . 6 ⊢ (𝑦 = 𝑣 → (∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
79	78	ralab 3640	. . . . 5 ⊢ (∀𝑣 ∈ {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ ∀𝑣(∃𝑘 ∈ 𝐵 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
80	73, 79	bitri 275	. . . 4 ⊢ (∀𝑣 ∈ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ ∀𝑣(∃𝑘 ∈ 𝐵 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
81	70, 80	sylibr 234	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣 ∈ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
82		ralunb 4138	. . 3 ⊢ (∀𝑣 ∈ ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (∀𝑣 ∈ {∪ 𝐾} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ∧ ∀𝑣 ∈ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
83	39, 81, 82	sylanbrc 584	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣 ∈ ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
84	5	toptopon 22896	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
85	31, 84	sylib 218	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
86		snex 5378	. . . 4 ⊢ {∪ 𝐾} ∈ V
87		fvex 6849	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐵)‘𝑘) ∈ V
88	87	abrexex 7910	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
89	88	rgenw 3056	. . . . . 6 ⊢ ∀𝑘 ∈ 𝐵 {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
90		abrexex2g 7912	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ∀𝑘 ∈ 𝐵 {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V) → {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V)
91	17, 89, 90	sylancl 587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V)
92	72, 91	eqeltrid 2841	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ V)
93		unexg 7692	. . . 4 ⊢ (({∪ 𝐾} ∈ V ∧ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ V) → ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))) ∈ V)
94	86, 92, 93	sylancr 588	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))) ∈ V)
95		eqid 2737	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
96	20, 95, 71	ptval2 23580	. . . 4 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → 𝐾 = (topGen‘(fi‘({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))))))
97	17, 19, 96	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 = (topGen‘(fi‘({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))))))
98		pttop 23561	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
99	17, 19, 98	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
100	20, 99	eqeltrid 2841	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ Top)
101	95	toptopon 22896	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
102	100, 101	sylib 218	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ (TopOn‘∪ 𝐾))
103	85, 94, 97, 102	subbascn 23233	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾 ∧ ∀𝑣 ∈ ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))(◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)))
104	26, 83, 103	mpbir2and 714	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5625  ran crn 5627   ↾ cres 5628   “ cima 5629   ∘ ccom 5630  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364  Xcixp 8840  ficfi 9318  topGenctg 17395  ∏_tcpt 17396  Topctop 22872  TopOnctopon 22889   Cn ccn 23203

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-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-1o 8400  df-2o 8401  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-fin 8892  df-fi 9319  df-topgen 17401  df-pt 17402  df-top 22873  df-topon 22890  df-bases 22925  df-cn 23206

This theorem is referenced by:  ptunhmeo  23787  tmdgsum  24074