Theorem xkoco2cn 23666

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 23274	. . . 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 7135	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7		eqid 2737	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8		eqid 2737	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
9		eqid 2737	. . . . . 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 23594	. . . . 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 2885	. . . 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 6073	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = ((𝐹 ∘ 𝑔) “ 𝑘))
16		imaco 6271	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝑔) “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘))
17	15, 16	eqtrdi 2793	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘)))
18	17	sseq1d 4015	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐹 ∘ 𝑔) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
19	18	elrab3 3693	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑆 = ∪ 𝑆
22		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑇 = ∪ 𝑇
23	21, 22	cnf 23254	. . . . . . . . . . . . . 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 6740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27		imassrn 6089	. . . . . . . . . . . . 13 ⊢ (𝑔 “ 𝑘) ⊆ ran 𝑔
28	7, 21	cnf 23254	. . . . . . . . . . . . . . 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 6744	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 ⊆ ∪ 𝑆)
31	27, 30	sstrid 3995	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ ∪ 𝑆)
32	25	fdmd 6746	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = ∪ 𝑆)
33	31, 32	sseqtrrd 4021	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ dom 𝐹)
34		funimass3 7074	. . . . . . . . . . 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 3443	. . . . . . . 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 23248	. . . . . . . . . . 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 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
44	43	elpwid 4609	. . . . . . . . 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 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
48		cnima 23273	. . . . . . . . . 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 23597	. . . . . . . 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 2841	. . . . . . 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 6074	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔))
54	53	mptpreima 6258	. . . . . . . . 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 2793	. . . . . . . 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 2826	. . . . . . 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 3213	. . . 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 3145	. 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 2737	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
63	62	xkotopon 23608	. . . 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 7464	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
66	65	pwex 5380	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
67	7, 8, 9	xkotf 23593	. . . . . 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 6743	. . . . . 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 5322	. . . 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 23249	. . . . 5 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
73	2, 72	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Top)
74	7, 8, 9	xkoval 23595	. . . 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 2737	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
77	76	xkotopon 23608	. . . 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 23262	. 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 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ficfi 9450   ↾_t crest 17465  topGenctg 17482  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  Compccmp 23394   ↑_ko cxko 23569

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-cmp 23395  df-xko 23571

This theorem is referenced by:  cnmptk1  23689