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Theorem xkococn 21684

Description: Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypothesis**
Ref	Expression
xkococn.1	⊢ 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓 ∘ 𝑔))

**Assertion**
Ref	Expression
xkococn	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) Cn (𝑇 ^_ko 𝑅)))

Distinct variable groups: 𝑓,𝑔,𝑅 𝑆,𝑓,𝑔 𝑇,𝑓,𝑔 Allowed substitution hints: 𝐹(𝑓,𝑔)

Proof of Theorem xkococn Dummy variables 𝑘 𝑎 𝑣 𝑥 𝑦 𝑧 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 756	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2		simprl 754	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3		cnco 21291	. . . . 5 ⊢ ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
4	1, 2, 3	syl2anc 573	. . . 4 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
5	4	ralrimivva 3120	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ∀𝑓 ∈ (𝑆 Cn 𝑇)∀𝑔 ∈ (𝑅 Cn 𝑆)(𝑓 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
6		xkococn.1	. . . 4 ⊢ 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓 ∘ 𝑔))
7	6	fmpt2 7387	. . 3 ⊢ (∀𝑓 ∈ (𝑆 Cn 𝑇)∀𝑔 ∈ (𝑅 Cn 𝑆)(𝑓 ∘ 𝑔) ∈ (𝑅 Cn 𝑇) ↔ 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
8	5, 7	sylib 208	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
9		eqid 2771	. . . . . . 7 ⊢ (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})
10	9	rnmpt2 6917	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}}
11	10	eleq2i 2842	. . . . 5 ⊢ (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}})
12		abid 2759	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} ↔ ∃𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})
13		oveq2 6801	. . . . . . 7 ⊢ (𝑦 = 𝑘 → (𝑅 ↾_t 𝑦) = (𝑅 ↾_t 𝑘))
14	13	eleq1d 2835	. . . . . 6 ⊢ (𝑦 = 𝑘 → ((𝑅 ↾_t 𝑦) ∈ Comp ↔ (𝑅 ↾_t 𝑘) ∈ Comp))
15	14	rexrab 3522	. . . . 5 ⊢ (∃𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑅((𝑅 ↾_t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
16	11, 12, 15	3bitri 286	. . . 4 ⊢ (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑅((𝑅 ↾_t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
17	8	ad2antrr 705	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
18		ffn 6185	. . . . . . . . . . . . 13 ⊢ (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19		elpreima 6480	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) → (𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ↔ (𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})))
20	17, 18, 19	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → (𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ↔ (𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})))
21		coeq1 5418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑎 → (𝑓 ∘ 𝑔) = (𝑎 ∘ 𝑔))
22		coeq2 5419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑏 → (𝑎 ∘ 𝑔) = (𝑎 ∘ 𝑏))
23		vex 3354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑎 ∈ V
24		vex 3354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑏 ∈ V
25	23, 24	coex 7265	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∘ 𝑏) ∈ V
26	21, 22, 6, 25	ovmpt2 6943	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎 ∘ 𝑏))
27	26	adantl 467	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → (𝑎𝐹𝑏) = (𝑎 ∘ 𝑏))
28	27	eleq1d 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
29		imaeq1 5602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑎 ∘ 𝑏) → (ℎ “ 𝑘) = ((𝑎 ∘ 𝑏) “ 𝑘))
30	29	sseq1d 3781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑎 ∘ 𝑏) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣))
31	30	elrab 3515	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ ((𝑎 ∘ 𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣))
32	31	simprbi 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)
33		simp2 1131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
34	33	ad3antrrr 709	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑆 ∈ 𝑛-Locally Comp)
35		elpwi 4307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝒫 ∪ 𝑅 → 𝑘 ⊆ ∪ 𝑅)
36	35	ad2antrl 707	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) → 𝑘 ⊆ ∪ 𝑅)
37	36	ad2antrr 705	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑘 ⊆ ∪ 𝑅)
38		simprr 756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) → (𝑅 ↾_t 𝑘) ∈ Comp)
39	38	ad2antrr 705	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → (𝑅 ↾_t 𝑘) ∈ Comp)
40		simplr 752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑣 ∈ 𝑇)
41		simprll 764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑎 ∈ (𝑆 Cn 𝑇))
42		simprlr 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑏 ∈ (𝑅 Cn 𝑆))
43		simprr 756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)
44	6, 34, 37, 39, 40, 41, 42, 43	xkococnlem 21683	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)) → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})))
45	44	expr 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → (((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣 → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
46	32, 45	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
47	28, 46	sylbid 230	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
48	47	ralrimivva 3120	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ∀𝑎 ∈ (𝑆 Cn 𝑇)∀𝑏 ∈ (𝑅 Cn 𝑆)((𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
49		fveq2 6332	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50		df-ov 6796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
51	49, 50	syl6eqr 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑦) = (𝑎𝐹𝑏))
52	51	eleq1d 2835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
53		eleq1 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ 𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
54	53	anbi1d 615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})) ↔ (⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
55	54	rexbidv 3200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})) ↔ ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
56	52, 55	imbi12d 333	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))) ↔ ((𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})))))
57	56	ralxp 5402	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))) ↔ ∀𝑎 ∈ (𝑆 Cn 𝑇)∀𝑏 ∈ (𝑅 Cn 𝑆)((𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
58	48, 57	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ∀𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
59	58	r19.21bi 3081	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) ∧ 𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))) → ((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
60	59	expimpd 441	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ((𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
61	20, 60	sylbid 230	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → (𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → ∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
62	61	ralrimiv 3114	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ∀𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})))
63		nllytop 21497	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64	63	3ad2ant2 1128	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65		simp3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66		xkotop 21612	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ Top)
67	64, 65, 66	syl2anc 573	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ Top)
68		simp1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69		xkotop 21612	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ Top)
70	68, 64, 69	syl2anc 573	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ Top)
71		txtop 21593	. . . . . . . . . . . . 13 ⊢ (((𝑇 ^_ko 𝑆) ∈ Top ∧ (𝑆 ^_ko 𝑅) ∈ Top) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top)
72	67, 70, 71	syl2anc 573	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top)
73	72	ad2antrr 705	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top)
74		eltop2 21000	. . . . . . . . . . 11 ⊢ (((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top → ((^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ↔ ∀𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
75	73, 74	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → ((^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ↔ ∀𝑦 ∈ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
76	62, 75	mpbird 247	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)))
77		imaeq2 5603	. . . . . . . . . 10 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
78	77	eleq1d 2835	. . . . . . . . 9 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ((^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ↔ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
79	76, 78	syl5ibrcom 237	. . . . . . . 8 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) ∧ 𝑣 ∈ 𝑇) → (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
80	79	rexlimdva 3179	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) → (∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
81	80	anassrs 458	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 ∪ 𝑅) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → (∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
82	81	expimpd 441	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 ∪ 𝑅) → (((𝑅 ↾_t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
83	82	rexlimdva 3179	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (∃𝑘 ∈ 𝒫 ∪ 𝑅((𝑅 ↾_t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑇 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
84	16, 83	syl5bi 232	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅))))
85	84	ralrimiv 3114	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)))
86		eqid 2771	. . . . . 6 ⊢ (𝑇 ^_ko 𝑆) = (𝑇 ^_ko 𝑆)
87	86	xkotopon 21624	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
88	64, 65, 87	syl2anc 573	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89		eqid 2771	. . . . . 6 ⊢ (𝑆 ^_ko 𝑅) = (𝑆 ^_ko 𝑅)
90	89	xkotopon 21624	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
91	68, 64, 90	syl2anc 573	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92		txtopon 21615	. . . 4 ⊢ (((𝑇 ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)) ∧ (𝑆 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆))) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
93	88, 91, 92	syl2anc 573	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
94		ovex 6823	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
95	94	pwex 4981	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
96		eqid 2771	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
97		eqid 2771	. . . . . . 7 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}
98	96, 97, 9	xkotf 21609	. . . . . 6 ⊢ (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}):({𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp} × 𝑇)⟶𝒫 (𝑅 Cn 𝑇)
99		frn 6193	. . . . . 6 ⊢ ((𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}):({𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp} × 𝑇)⟶𝒫 (𝑅 Cn 𝑇) → ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇))
100	98, 99	ax-mp 5	. . . . 5 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇)
101	95, 100	ssexi 4937	. . . 4 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V
102	101	a1i 11	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V)
103	96, 97, 9	xkoval 21611	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
104	103	3adant2 1125	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
105		eqid 2771	. . . . 5 ⊢ (𝑇 ^_ko 𝑅) = (𝑇 ^_ko 𝑅)
106	105	xkotopon 21624	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
107	106	3adant2 1125	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
108	93, 102, 104, 107	subbascn 21279	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝐹 ∈ (((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) Cn (𝑇 ^_ko 𝑅)) ↔ (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) ∧ ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑥) ∈ ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)))))
109	8, 85, 108	mpbir2and 692	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) Cn (𝑇 ^_ko 𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 {cab 2757 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3351 ⊆ wss 3723 𝒫 cpw 4297 ⟨cop 4322 ∪ cuni 4574 × cxp 5247 ^◡ccnv 5248 ran crn 5250 “ cima 5252 ∘ ccom 5253 Fn wfn 6026 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 ficfi 8472 ↾_t crest 16289 topGenctg 16306 Topctop 20918 TopOnctopon 20935 Cn ccn 21249 Compccmp 21410 𝑛-Locally cnlly 21489 ×_t ctx 21584 ^_ko cxko 21585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-fin 8113 df-fi 8473 df-rest 16291 df-topgen 16312 df-top 20919 df-topon 20936 df-bases 20971 df-ntr 21045 df-nei 21123 df-cn 21252 df-cmp 21411 df-nlly 21491 df-tx 21586 df-xko 21587

This theorem is referenced by: cnmptkk 21707 xkofvcn 21708 symgtgp 22125

W3C validator