Theorem xkoco2cn 23885

Description: If 𝐹 is a continuous function, then 𝑔 ↦ 𝐹 ∘ 𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypotheses

Ref	Expression
xkoco2cn.r	⊢ (𝜑 → 𝑅 ∈ Top)
xkoco2cn.f	⊢ (𝜑 → 𝐹 ∈ (𝑆 Cn 𝑇))

Assertion

Ref	Expression
xkoco2cn	⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))

Distinct variable groups:   𝜑,𝑔   𝑅,𝑔   𝑆,𝑔   𝑇,𝑔   𝑔,𝐹

Proof of Theorem xkoco2cn

Dummy variables 𝑘 𝑣 𝑥 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2		xkoco2cn.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑆 Cn 𝑇))
3	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4		cnco 23492	. . . 4 ⊢ ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
5	1, 3, 4	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
6	5	fmpttd 7111	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7		eqid 2762	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8		eqid 2762	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
9		eqid 2762	. . . . . 6 ⊢ (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	7, 8, 9	xkobval 23813	. . . . 5 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑇 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})}
11	10	eqabri 2904	. . . 4 ⊢ (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑇 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
12		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
13	2	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
14	12, 13, 4	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
15		imaeq1 6055	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = ((𝐹 ∘ 𝑔) “ 𝑘))
16		imaco 6251	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝑔) “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘))
17	15, 16	eqtrdi 2813	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘)))
18	17	sseq1d 3965	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐹 ∘ 𝑔) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
19	18	elrab3 3649	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇) → ((𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
20	14, 19	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
21		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑆 = ∪ 𝑆
22		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑇 = ∪ 𝑇
23	21, 22	cnf 23472	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹:∪ 𝑆⟶∪ 𝑇)
24	2, 23	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
25	24	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
26	25	ffund 6711	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27		imassrn 6071	. . . . . . . . . . . . 13 ⊢ (𝑔 “ 𝑘) ⊆ ran 𝑔
28	7, 21	cnf 23472	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔:∪ 𝑅⟶∪ 𝑆)
29	12, 28	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔:∪ 𝑅⟶∪ 𝑆)
30	29	frnd 6715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 ⊆ ∪ 𝑆)
31	27, 30	sstrid 3945	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ ∪ 𝑆)
32	25	fdmd 6717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = ∪ 𝑆)
33	31, 32	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ dom 𝐹)
34		funimass3 7050	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ (𝑔 “ 𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣 ↔ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)))
35	26, 33, 34	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣 ↔ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)))
36	20, 35	bitrd 282	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)))
37	36	rabbidva 3420	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)})
38		xkoco2cn.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Top)
39	38	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
40		cntop1 23466	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
41	2, 40	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Top)
42	41	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
43		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
44	43	elpwid 4569	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑅)
45		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑅 ↾t 𝑘) ∈ Comp)
46	2	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
47		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
48		cnima 23491	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ 𝑇) → (◡𝐹 “ 𝑣) ∈ 𝑆)
49	46, 47, 48	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (◡𝐹 “ 𝑣) ∈ 𝑆)
50	7, 39, 42, 44, 45, 49	xkoopn 23816	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)} ∈ (𝑆 ↑ko 𝑅))
51	37, 50	eqeltrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} ∈ (𝑆 ↑ko 𝑅))
52		imaeq2 6056	. . . . . . . . 9 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) = (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
53		eqid 2762	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔))
54	53	mptpreima 6238	. . . . . . . . 9 ⊢ (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}}
55	52, 54	eqtrdi 2813	. . . . . . . 8 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}})
56	55	eleq1d 2847	. . . . . . 7 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ((◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅) ↔ {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} ∈ (𝑆 ↑ko 𝑅)))
57	51, 56	syl5ibrcom 250	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅)))
58	57	expimpd 459	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) → (((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅)))
59	58	rexlimdvva 3221	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑇 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅)))
60	11, 59	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅)))
61	60	ralrimiv 3155	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})(◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅))
62		eqid 2762	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
63	62	xkotopon 23827	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
64	38, 41, 63	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
65		ovex 7449	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
66	65	pwex 5349	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
67	7, 8, 9	xkotf 23812	. . . . . 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)
68		frn 6714	. . . . . 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))
69	67, 68	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)
70	66, 69	ssexi 5291	. . . 4 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V
71	70	a1i 11	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V)
72		cntop2 23467	. . . . 5 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
73	2, 72	syl 18	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Top)
74	7, 8, 9	xkoval 23814	. . . 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}))))
75	38, 73, 74	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))))
76		eqid 2762	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
77	76	xkotopon 23827	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
78	38, 73, 77	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
79	64, 71, 75, 78	subbascn 23480	. 2 ⊢ (𝜑 → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)) ↔ ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)):(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})(◡(𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) “ 𝑥) ∈ (𝑆 ↑ko 𝑅))))
80	6, 61, 79	mpbir2and 726	1 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5190   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662   ∘ ccom 5663  Fun wfun 6531  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  ficfi 9383   ↾_t crest 17509  topGenctg 17526  Topctop 23119  TopOnctopon 23136   Cn ccn 23450  Compccmp 23612   ↑_ko cxko 23788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-1o 8458  df-2o 8459  df-map 8831  df-en 8956  df-dom 8957  df-fin 8959  df-fi 9384  df-rest 17511  df-topgen 17532  df-top 23120  df-topon 23137  df-bases 23172  df-cn 23453  df-cmp 23613  df-xko 23790

This theorem is used by:  cnmptk1  23908