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Theorem xkococn 21677

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 780	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2		simprl 778	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3		cnco 21284	. . . . 5 ⊢ ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
4	1, 2, 3	syl2anc 575	. . . 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 3159	. . 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 7470	. . 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 209	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
9		eqid 2806	. . . . . . 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 7000	. . . . . 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 2877	. . . . 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 2794	. . . . 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 6882	. . . . . . 7 ⊢ (𝑦 = 𝑘 → (𝑅 ↾_t 𝑦) = (𝑅 ↾_t 𝑘))
14	13	eleq1d 2870	. . . . . 6 ⊢ (𝑦 = 𝑘 → ((𝑅 ↾_t 𝑦) ∈ Comp ↔ (𝑅 ↾_t 𝑘) ∈ Comp))
15	14	rexrab 3566	. . . . 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 288	. . . 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 708	. . . . . . . . . . . . 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 6256	. . . . . . . . . . . . 13 ⊢ (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19		elpreima 6559	. . . . . . . . . . . . 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 5481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑎 → (𝑓 ∘ 𝑔) = (𝑎 ∘ 𝑔))
22		coeq2 5482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑏 → (𝑎 ∘ 𝑔) = (𝑎 ∘ 𝑏))
23		vex 3394	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑎 ∈ V
24		vex 3394	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑏 ∈ V
25	23, 24	coex 7348	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∘ 𝑏) ∈ V
26	21, 22, 6, 25	ovmpt2 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎 ∘ 𝑏))
27	26	adantl 469	. . . . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . . . 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 5671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑎 ∘ 𝑏) → (ℎ “ 𝑘) = ((𝑎 ∘ 𝑏) “ 𝑘))
30	29	sseq1d 3829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑎 ∘ 𝑏) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣))
31	30	elrab 3559	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ ((𝑎 ∘ 𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣))
32	31	simprbi 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∘ 𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ((𝑎 ∘ 𝑏) “ 𝑘) ⊆ 𝑣)
33		simp2 1160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
34	33	ad3antrrr 712	. . . . . . . . . . . . . . . . . . . 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 4361	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝒫 ∪ 𝑅 → 𝑘 ⊆ ∪ 𝑅)
36	35	ad2antrl 710	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) → 𝑘 ⊆ ∪ 𝑅)
37	36	ad2antrr 708	. . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ (𝑅 ↾_t 𝑘) ∈ Comp)) → (𝑅 ↾_t 𝑘) ∈ Comp)
39	38	ad2antrr 708	. . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . 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 21676	. . . . . . . . . . . . . . . . . . 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 446	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . 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 6408	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50		df-ov 6877	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
51	49, 50	syl6eqr 2858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑦) = (𝑎𝐹𝑏))
52	51	eleq1d 2870	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑦) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
53		eleq1 2873	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ 𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
54	53	anbi1d 617	. . . . . . . . . . . . . . . . . 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 3240	. . . . . . . . . . . . . . . . 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 335	. . . . . . . . . . . . . . . 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 5465	. . . . . . . . . . . . . . 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 225	. . . . . . . . . . . . . 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 3120	. . . . . . . . . . . . 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 443	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 3153	. . . . . . . . . 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 21490	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64	63	3ad2ant2 1157	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65		simp3 1161	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66		xkotop 21605	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ Top)
67	64, 65, 66	syl2anc 575	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ Top)
68		simp1 1159	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69		xkotop 21605	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ Top)
70	68, 64, 69	syl2anc 575	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ Top)
71		txtop 21586	. . . . . . . . . . . . 13 ⊢ (((𝑇 ^_ko 𝑆) ∈ Top ∧ (𝑆 ^_ko 𝑅) ∈ Top) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top)
72	67, 70, 71	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^_ko 𝑆) ×_t (𝑆 ^_ko 𝑅)) ∈ Top)
73	72	ad2antrr 708	. . . . . . . . . . 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 20993	. . . . . . . . . . 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 248	. . . . . . . . 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 5672	. . . . . . . . . 10 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
78	77	eleq1d 2870	. . . . . . . . 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 238	. . . . . . . 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 3219	. . . . . . 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 455	. . . . . 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 443	. . . . 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 3219	. . . 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 233	. . 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 3153	. 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 2806	. . . . . 6 ⊢ (𝑇 ^_ko 𝑆) = (𝑇 ^_ko 𝑆)
87	86	xkotopon 21617	. . . . 5 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
88	64, 65, 87	syl2anc 575	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89		eqid 2806	. . . . . 6 ⊢ (𝑆 ^_ko 𝑅) = (𝑆 ^_ko 𝑅)
90	89	xkotopon 21617	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
91	68, 64, 90	syl2anc 575	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92		txtopon 21608	. . . 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 575	. . 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 6906	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
95	94	pwex 5050	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
96		eqid 2806	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
97		eqid 2806	. . . . . . 7 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑦) ∈ Comp}
98	96, 97, 9	xkotf 21602	. . . . . 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 6262	. . . . . 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 4998	. . . 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 21604	. . . 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 1154	. . 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 2806	. . . . 5 ⊢ (𝑇 ^_ko 𝑅) = (𝑇 ^_ko 𝑅)
106	105	xkotopon 21617	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
107	106	3adant2 1154	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
108	93, 102, 104, 107	subbascn 21272	. 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 695	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 197 ∧ wa 384 ∧ w3a 1100 = wceq 1637 ∈ wcel 2156 {cab 2792 ∀wral 3096 ∃wrex 3097 {crab 3100 Vcvv 3391 ⊆ wss 3769 𝒫 cpw 4351 ⟨cop 4376 ∪ cuni 4630 × cxp 5309 ^◡ccnv 5310 ran crn 5312 “ cima 5314 ∘ ccom 5315 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 (class class class)co 6874 ↦ cmpt2 6876 ficfi 8555 ↾_t crest 16286 topGenctg 16303 Topctop 20911 TopOnctopon 20928 Cn ccn 21242 Compccmp 21403 𝑛-Locally cnlly 21482 ×_t ctx 21577 ^_ko cxko 21578

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7179

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-ral 3101 df-rex 3102 df-reu 3103 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-tp 4375 df-op 4377 df-uni 4631 df-int 4670 df-iun 4714 df-iin 4715 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5893 df-ord 5939 df-on 5940 df-lim 5941 df-suc 5942 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6877 df-oprab 6878 df-mpt2 6879 df-om 7296 df-1st 7398 df-2nd 7399 df-wrecs 7642 df-recs 7704 df-rdg 7742 df-1o 7796 df-oadd 7800 df-er 7979 df-map 8094 df-en 8193 df-dom 8194 df-fin 8196 df-fi 8556 df-rest 16288 df-topgen 16309 df-top 20912 df-topon 20929 df-bases 20964 df-ntr 21038 df-nei 21116 df-cn 21245 df-cmp 21404 df-nlly 21484 df-tx 21579 df-xko 21580

This theorem is referenced by: cnmptkk 21700 xkofvcn 21701 symgtgp 22118

W3C validator