Theorem ptrescn 21442

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 1066	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ⊆ 𝐴)
2		ptrescn.2	. . . . . . . . . 10 ⊢ 𝐽 = (∏t‘𝐹)
3	2	ptuni 21397	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
4	3	3adant3 1081	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐽)
5		ptrescn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	syl6eqr 2674	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
7	6	eleq2d 2687	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ 𝑥 ∈ 𝑋))
8	7	biimpar 502	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
9		resixp 7943	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
10	1, 8, 9	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
11		ixpeq2 7922	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
12		fvres 6207	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
13	12	unieqd 4446	. . . . . . 7 ⊢ (𝑘 ∈ 𝐵 → ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘))
14	11, 13	mprg 2926	. . . . . 6 ⊢ X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘)
15		ssexg 4804	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
16	15	ancoms 469	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
17	16	3adant2 1080	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
18		fssres 6070	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
19	18	3adant1 1079	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶Top)
20		ptrescn.3	. . . . . . . 8 ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐵))
21	20	ptuni 21397	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
22	17, 19, 21	syl2anc 693	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐾)
23	14, 22	syl5eqr 2670	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
24	23	adantr 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐾)
25	10, 24	eleqtrd 2703	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
26		eqid 2622	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))
27	25, 26	fmptd 6385	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾)
28		fimacnv 6347	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)):𝑋⟶∪ 𝐾 → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) = 𝑋)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) = 𝑋)
30		pttop 21385	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
31	2, 30	syl5eqel 2705	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 ∈ Top)
32	31	3adant3 1081	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
33	5	topopn 20711	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
34	32, 33	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝑋 ∈ 𝐽)
35	29, 34	eqeltrd 2701	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽)
36		elsni 4194	. . . . . . 7 ⊢ (𝑣 ∈ {∪ 𝐾} → 𝑣 = ∪ 𝐾)
37	36	imaeq2d 5466	. . . . . 6 ⊢ (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾))
38	37	eleq1d 2686	. . . . 5 ⊢ (𝑣 ∈ {∪ 𝐾} → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ ∪ 𝐾) ∈ 𝐽))
39	35, 38	syl5ibrcom 237	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ {∪ 𝐾} → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽))
40	39	ralrimiv 2965	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ∀𝑣 ∈ {∪ 𝐾} (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽)
41		imaco 5640	. . . . . . . . 9 ⊢ ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘))) “ 𝑢) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))
42		cnvco 5308	. . . . . . . . . . 11 ⊢ ◡((𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) ∘ (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵))) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∘ ◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
43	25	adantlr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) ∧ 𝑥 ∈ 𝑋) → (𝑥 ↾ 𝐵) ∈ ∪ 𝐾)
44		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)))
45		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) = (𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)))
46		fveq1 6190	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥 ↾ 𝐵) → (𝑧‘𝑘) = ((𝑥 ↾ 𝐵)‘𝑘))
47	43, 44, 45, 46	fmptco 6396	. . . . . . . . . . . . 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)))
48		fvres 6207	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐵 → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
49	48	ad2antrl 764	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((𝑥 ↾ 𝐵)‘𝑘) = (𝑥‘𝑘))
50	49	mpteq2dv 4745	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ ((𝑥 ↾ 𝐵)‘𝑘)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)))
51	47, 50	eqtrd 2656	. . . . . . . . . . . 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)))
52	51	cnveqd 5298	. . . . . . . . . . 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)))
53	42, 52	syl5eqr 2670	. . . . . . . . . 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)))
54	53	imaeq1d 5465	. . . . . . . . 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))
55	41, 54	syl5eqr 2670	. . . . . . . 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))
56		simpl1 1064	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐴 ∈ 𝑉)
57		simpl2 1065	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐹:𝐴⟶Top)
58		simpl3 1066	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝐵 ⊆ 𝐴)
59		simprl 794	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐵)
60	58, 59	sseldd 3604	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑘 ∈ 𝐴)
61	5, 2	ptpjcn 21414	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
62	56, 57, 60, 61	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)))
63		simprr 796	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑢 ∈ (𝐹‘𝑘))
64		cnima 21069	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑢 ∈ (𝐹‘𝑘)) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
65	62, 63, 64	syl2anc 693	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥‘𝑘)) “ 𝑢) ∈ 𝐽)
66	55, 65	eqeltrd 2701	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) ∧ (𝑘 ∈ 𝐵 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ 𝐽)
67		imaeq2 5462	. . . . . . . 8 ⊢ (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) = (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
68	67	eleq1d 2686	. . . . . . 7 ⊢ (𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) → ((◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ 𝑣) ∈ 𝐽 ↔ (◡(𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) “ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ 𝐽))
69	66, 68	syl5ibrcom 237	. . . . . 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))
70	69	rexlimdvva 3038	. . . . 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))
71	70	alrimiv 1855	. . . 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))
72		eqid 2622	. . . . . . 7 ⊢ (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))
73	72	rnmpt2 6770	. . . . . 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)}
74	73	raleqi 3142	. . . . 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)
75	12	rexeqdv 3145	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → (∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
76		eqeq1 2626	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ 𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
77	76	rexbidv 3052	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (∃𝑢 ∈ (𝐹‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
78	75, 77	sylan9bbr 737	. . . . . . 7 ⊢ ((𝑦 = 𝑣 ∧ 𝑘 ∈ 𝐵) → (∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢) ↔ ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)))
79	78	rexbidva 3049	. . . . . 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)))
80	79	ralab 3367	. . . . 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))
81	74, 80	bitri 264	. . . 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))
82	71, 81	sylibr 224	. . 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)
83		ralunb 3794	. . 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))
84	40, 82, 83	sylanbrc 698	. 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)
85	5	toptopon 20722	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
86	32, 85	sylib 208	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
87		snex 4908	. . . 4 ⊢ {∪ 𝐾} ∈ V
88		fvex 6201	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐵)‘𝑘) ∈ V
89	88	abrexex 7141	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
90	89	rgenw 2924	. . . . . 6 ⊢ ∀𝑘 ∈ 𝐵 {𝑦 ∣ ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V
91		abrexex2g 7144	. . . . . 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)
92	17, 90, 91	sylancl 694	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → {𝑦 ∣ ∃𝑘 ∈ 𝐵 ∃𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘)𝑦 = (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)} ∈ V)
93	73, 92	syl5eqel 2705	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢)) ∈ V)
94		unexg 6959	. . . 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)
95	87, 93, 94	sylancr 695	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → ({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))) ∈ V)
96		eqid 2622	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
97	20, 96, 72	ptval2 21404	. . . 4 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → 𝐾 = (topGen‘(fi‘({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))))))
98	17, 19, 97	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 = (topGen‘(fi‘({∪ 𝐾} ∪ ran (𝑘 ∈ 𝐵, 𝑢 ∈ ((𝐹 ↾ 𝐵)‘𝑘) ↦ (◡(𝑧 ∈ ∪ 𝐾 ↦ (𝑧‘𝑘)) “ 𝑢))))))
99		pttop 21385	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
100	17, 19, 99	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
101	20, 100	syl5eqel 2705	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ Top)
102	96	toptopon 20722	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
103	101, 102	sylib 208	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → 𝐾 ∈ (TopOn‘∪ 𝐾))
104	86, 95, 98, 103	subbascn 21058	. 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)))
105	27, 84, 104	mpbir2and 957	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Xcixp 7908  ficfi 8316  topGenctg 16098  ∏_tcpt 16099  Topctop 20698  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031

This theorem is referenced by:  ptunhmeo  21611  tmdgsum  21899