Theorem xkoco2cn 22266

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 487	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2		xkoco2cn.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑆 Cn 𝑇))
3	2	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4		cnco 21874	. . . 4 ⊢ ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
5	1, 3, 4	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
6	5	fmpttd 6879	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7		eqid 2821	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
8		eqid 2821	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
9		eqid 2821	. . . . . 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 22194	. . . . 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	abeq2i 2948	. . . 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 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
13	2	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
14	12, 13, 4	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹 ∘ 𝑔) ∈ (𝑅 Cn 𝑇))
15		imaeq1 5924	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = ((𝐹 ∘ 𝑔) “ 𝑘))
16		imaco 6104	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∘ 𝑔) “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘))
17	15, 16	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐹 ∘ 𝑔) → (ℎ “ 𝑘) = (𝐹 “ (𝑔 “ 𝑘)))
18	17	sseq1d 3998	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐹 ∘ 𝑔) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣))
19	18	elrab3 3681	. . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑆 = ∪ 𝑆
22		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑇 = ∪ 𝑇
23	21, 22	cnf 21854	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹:∪ 𝑆⟶∪ 𝑇)
24	2, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
25	24	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
26	25	ffund 6518	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27		imassrn 5940	. . . . . . . . . . . . 13 ⊢ (𝑔 “ 𝑘) ⊆ ran 𝑔
28	7, 21	cnf 21854	. . . . . . . . . . . . . . 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 6521	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 ⊆ ∪ 𝑆)
31	27, 30	sstrid 3978	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ ∪ 𝑆)
32	25	fdmd 6523	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = ∪ 𝑆)
33	31, 32	sseqtrrd 4008	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔 “ 𝑘) ⊆ dom 𝐹)
34		funimass3 6824	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ (𝑔 “ 𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔 “ 𝑘)) ⊆ 𝑣 ↔ (𝑔 “ 𝑘) ⊆ (◡𝐹 “ 𝑣)))
35	26, 33, 34	syl2anc 586	. . . . . . . . . 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 281	. . . . . . . . 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 3478	. . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
40		cntop1 21848	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
41	2, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Top)
42	41	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
43		simplrl 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
44	43	elpwid 4550	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑅)
45		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑅 ↾t 𝑘) ∈ Comp)
46	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
47		simplrr 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
48		cnima 21873	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ 𝑇) → (◡𝐹 “ 𝑣) ∈ 𝑆)
49	46, 47, 48	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (◡𝐹 “ 𝑣) ∈ 𝑆)
50	7, 39, 42, 44, 45, 49	xkoopn 22197	. . . . . . . 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 2913	. . . . . . 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 5925	. . . . . . . . 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 2821	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔))
54	53	mptpreima 6092	. . . . . . . . 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	syl6eq 2872	. . . . . . . 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 2897	. . . . . . 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 249	. . . . . 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 456	. . . . 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 3294	. . . 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	syl5bi 244	. . 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 3181	. 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 2821	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
63	62	xkotopon 22208	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
64	38, 41, 63	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
65		ovex 7189	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
66	65	pwex 5281	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
67	7, 8, 9	xkotf 22193	. . . . . 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 6520	. . . . . 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 5226	. . . 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 21849	. . . . 5 ⊢ (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
73	2, 72	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Top)
74	7, 8, 9	xkoval 22195	. . . 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 586	. . 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 2821	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
77	76	xkotopon 22208	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
78	38, 73, 77	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
79	64, 71, 75, 78	subbascn 21862	. 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 711	1 ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838   ↦ cmpt 5146   × cxp 5553  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   “ cima 5558   ∘ ccom 5559  Fun wfun 6349  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  ficfi 8874   ↾_t crest 16694  topGenctg 16711  Topctop 21501  TopOnctopon 21518   Cn ccn 21832  Compccmp 21994   ↑_ko cxko 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-fin 8513  df-fi 8875  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cn 21835  df-cmp 21995  df-xko 22171

This theorem is referenced by:  cnmptk1  22289