Theorem ptrescn 23626

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 1190	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ⊆ 𝐴)
2		ptrescn.2	. . . . . . . . . 10 ⊢ 𝐽 = (∏t‘𝐹)
3	2	ptuni 23581	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1129	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5		ptrescn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2783	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
7	6	eleq2d 2811	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ∈ 𝑋))
8	7	biimpar 476	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
9		resixp 8961	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
10	1, 8, 9	syl2anc 582	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
11		ixpeq2 8939	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
12		fvres 6919	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
13	12	unieqd 4925	. . . . . . 7 ⊢ (𝑘 ∈ 𝐵 → ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘))
14	11, 13	mprg 3056	. . . . . 6 ⊢ X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘)
15		ssexg 5327	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
16	15	ancoms 457	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
17	16	3adant2 1128	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
18		fssres 6767	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
19	18	3adant1 1127	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
20		ptrescn.3	. . . . . . . 8 ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐵))
21	20	ptuni 23581	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
22	17, 19, 21	syl2anc 582	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
23	14, 22	eqtr3id 2779	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
24	23	adantr 479	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
25	10, 24	eleqtrd 2827	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
26	25	fmpttd 7128	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾)
27		fimacnv 6749	. . . . . . 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 23569	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
30	2, 29	eqeltrid 2829	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 ∈ Top)
31	30	3adant3 1129	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
32	5	topopn 22891	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝑋 ∈ 𝐽)
34	28, 33	eqeltrd 2825	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽)
35		elsni 4649	. . . . . . 7 ⊢ (𝑣 ∈ {∪ 𝐾} → 𝑣 = ∪ 𝐾)
36	35	imaeq2d 6068	. . . . . 6 ⊢ (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾))
37	36	eleq1d 2810	. . . . 5 ⊢ (𝑣 ∈ {∪ 𝐾} → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽))
38	34, 37	syl5ibrcom 246	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
39	38	ralrimiv 3134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣 ∈ {∪ 𝐾} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
40		imaco 6261	. . . . . . . . 9 ⊢ ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘))) “ 𝑢) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))
41		cnvco 5891	. . . . . . . . . . 11 ⊢ ◡((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
42	25	adantlr 713	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
43		eqidd 2726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)))
44		eqidd 2726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) = (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
45		fveq1 6899	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 ↾ 𝐵) → (𝑧‘𝑘) = ((𝑥 ↾ 𝐵)‘𝑘))
46	42, 43, 44, 45	fmptco 7142	. . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐵 → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
48	47	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
49	48	mpteq2dv 5254	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . 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 5881	. . . . . . . . . . 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 2779	. . . . . . . . . 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 6067	. . . . . . . . 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 2779	. . . . . . . 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 1188	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐴 ∈ 𝑉)
56		simpl2 1189	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐹:𝐴⟶Top)
57		simpl3 1190	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐵 ⊆ 𝐴)
58		simprl 769	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐵)
59	57, 58	sseldd 3979	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐴)
60	5, 2	ptpjcn 23598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
61	55, 56, 59, 60	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
62		simprr 771	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑢 ∈ (𝐹‘𝑘))
63		cnima 23252	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑢 ∈ (𝐹‘𝑘)) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
64	61, 62, 63	syl2anc 582	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
65	54, 64	eqeltrd 2825	. . . . . . 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 6064	. . . . . . . 8 ⊢ (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
67	66	eleq1d 2810	. . . . . . 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 246	. . . . . 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 3201	. . . . 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 1922	. . . 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 2725	. . . . . . 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 7558	. . . . . 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 3312	. . . . 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 3315	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → (∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
75		eqeq1 2729	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ 𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
76	75	rexbidv 3168	. . . . . . . 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 509	. . . . . . 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 3166	. . . . . 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 3684	. . . . 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 274	. . . 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 233	. . 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 4191	. . 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 581	. 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 22902	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
85	31, 84	sylib 217	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
86		snex 5436	. . . 4 ⊢ {∪ 𝐾} ∈ V
87		fvex 6913	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐵)‘𝑘) ∈ V
88	87	abrexex 7975	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
89	88	rgenw 3054	. . . . . 6 ⊢ ∀𝑘 ∈ 𝐵 {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
90		abrexex2g 7977	. . . . . 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 584	. . . . 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 2829	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ V)
93		unexg 7756	. . . 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 585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))) ∈ V)
95		eqid 2725	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
96	20, 95, 71	ptval2 23588	. . . 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 582	. . 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 23569	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
99	17, 19, 98	syl2anc 582	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
100	20, 99	eqeltrid 2829	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ Top)
101	95	toptopon 22902	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
102	100, 101	sylib 217	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ (TopOn‘∪ 𝐾))
103	85, 94, 97, 102	subbascn 23241	. 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 711	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ∪ cun 3944   ⊆ wss 3946  {csn 4632  ∪ cuni 4912   ↦ cmpt 5235  ^◡ccnv 5680  ran crn 5682   ↾ cres 5683   “ cima 5684   ∘ ccom 5685  ⟶wf 6549  ‘cfv 6553  (class class class)co 7423   ∈ cmpo 7425  Xcixp 8925  ficfi 9449  topGenctg 17447  ∏_tcpt 17448  Topctop 22878  TopOnctopon 22895   Cn ccn 23211

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-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  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-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-1o 8495  df-2o 8496  df-er 8733  df-map 8856  df-ixp 8926  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9450  df-topgen 17453  df-pt 17454  df-top 22879  df-topon 22896  df-bases 22932  df-cn 23214

This theorem is referenced by:  ptunhmeo  23795  tmdgsum  24082