Theorem xkoco2cn 23545

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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2		xkoco2cn.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑆 Cn 𝑇))
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4		cnco 23153	. . . 4 ⊢ ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
5	1, 3, 4	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
6	5	fmpttd 7087	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7		eqid 2729	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8		eqid 2729	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
9		eqid 2729	. . . . . 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 23473	. . . . 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 2871	. . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
13	2	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
14	12, 13, 4	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
15		imaeq1 6026	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = ((𝐹 ∘ 𝑔) “ 𝑘))
16		imaco 6224	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝑔) “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘))
17	15, 16	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘)))
18	17	sseq1d 3978	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐹 ∘ 𝑔) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
19	18	elrab3 3660	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇) → ((𝐹 ∘ 𝑔) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
20	14, 19	syl 17	. . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑆 = ∪ 𝑆
22		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑇 = ∪ 𝑇
23	21, 22	cnf 23133	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹:∪ 𝑆⟶∪ 𝑇)
24	2, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
25	24	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
26	25	ffund 6692	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27		imassrn 6042	. . . . . . . . . . . . 13 ⊢ (𝑔 “ 𝑘) ⊆ ran 𝑔
28	7, 21	cnf 23133	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔:∪ 𝑅⟶∪ 𝑆)
29	12, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔:∪ 𝑅⟶∪ 𝑆)
30	29	frnd 6696	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 ⊆ ∪ 𝑆)
31	27, 30	sstrid 3958	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ ∪ 𝑆)
32	25	fdmd 6698	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = ∪ 𝑆)
33	31, 32	sseqtrrd 3984	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ dom 𝐹)
34		funimass3 7026	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ (𝑔 “ 𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣 ↔ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)))
35	26, 33, 34	syl2anc 584	. . . . . . . . . 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 279	. . . . . . . . 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 3412	. . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
40		cntop1 23127	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
41	2, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Top)
42	41	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
43		simplrl 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
44	43	elpwid 4572	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑅)
45		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑅 ↾t 𝑘) ∈ Comp)
46	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
47		simplrr 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
48		cnima 23152	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ 𝑇) → (◡𝐹 “ 𝑣) ∈ 𝑆)
49	46, 47, 48	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (◡𝐹 “ 𝑣) ∈ 𝑆)
50	7, 39, 42, 44, 45, 49	xkoopn 23476	. . . . . . . 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 2828	. . . . . . 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 6027	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔))
54	53	mptpreima 6211	. . . . . . . . 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 2780	. . . . . . . 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 2813	. . . . . . 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 247	. . . . . 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 453	. . . . 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 3194	. . . 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 242	. . 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 3124	. 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 2729	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
63	62	xkotopon 23487	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
64	38, 41, 63	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
65		ovex 7420	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
66	65	pwex 5335	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
67	7, 8, 9	xkotf 23472	. . . . . 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 6695	. . . . . 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 5277	. . . 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 23128	. . . . 5 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
73	2, 72	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Top)
74	7, 8, 9	xkoval 23474	. . . 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 584	. . 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 2729	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
77	76	xkotopon 23487	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
78	38, 73, 77	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
79	64, 71, 75, 78	subbascn 23141	. 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 713	1 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641   ∘ ccom 5642  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ficfi 9361   ↾_t crest 17383  topGenctg 17400  Topctop 22780  TopOnctopon 22797   Cn ccn 23111  Compccmp 23273   ↑_ko cxko 23448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-2o 8435  df-map 8801  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cn 23114  df-cmp 23274  df-xko 23450

This theorem is referenced by:  cnmptk1  23568