Theorem xkoco1cn 23775

Description: If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 23776 independently of the more general xkococn 23778 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)

Hypotheses

Ref	Expression
xkoco1cn.t	⊢ (𝜑 → 𝑇 ∈ Top)
xkoco1cn.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 Cn 𝑆))

Assertion

Ref	Expression
xkoco1cn	⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))

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

Proof of Theorem xkoco1cn

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

Step	Hyp	Ref	Expression
1		xkoco1cn.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 Cn 𝑆))
2		cnco 23384	. . . 4 ⊢ ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
3	1, 2	sylan 591	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
4	3	fmpttd 7100	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
5		eqid 2765	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2765	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
7		eqid 2765	. . . . . 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})
8	5, 6, 7	xkobval 23704	. . . . 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})}
9	8	eqabri 2907	. . . 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}))
10	1	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
11	10, 2	sylan 591	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
12		imaeq1 6048	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑔 ∘ 𝐹) → (ℎ “ 𝑘) = ((𝑔 ∘ 𝐹) “ 𝑘))
13		imaco 6242	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∘ 𝐹) “ 𝑘) = (𝑔 “ (𝐹 “ 𝑘))
14	12, 13	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑔 ∘ 𝐹) → (ℎ “ 𝑘) = (𝑔 “ (𝐹 “ 𝑘)))
15	14	sseq1d 3970	. . . . . . . . . . 11 ⊢ (ℎ = (𝑔 ∘ 𝐹) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣))
16	15	elrab3 3654	. . . . . . . . . 10 ⊢ ((𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇) → ((𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣))
17	11, 16	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → ((𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣))
18	17	rabbidva 3423	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣})
19		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝑆 = ∪ 𝑆
20		cntop2 23359	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
21	1, 20	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Top)
22	21	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
23		xkoco1cn.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ Top)
24	23	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
25		imassrn 6064	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑘) ⊆ ran 𝐹
26	5, 19	cnf 23364	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹:∪ 𝑅⟶∪ 𝑆)
27		frn 6703	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝑅⟶∪ 𝑆 → ran 𝐹 ⊆ ∪ 𝑆)
28	10, 26, 27	3syl 19	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → ran 𝐹 ⊆ ∪ 𝑆)
29	25, 28	sstrid 3950	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝐹 “ 𝑘) ⊆ ∪ 𝑆)
30		imacmp 23515	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑆 ↾t (𝐹 “ 𝑘)) ∈ Comp)
31	10, 30	sylancom 599	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑆 ↾t (𝐹 “ 𝑘)) ∈ Comp)
32		simplrr 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
33	19, 22, 24, 29, 31, 32	xkoopn 23707	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣} ∈ (𝑇 ↑ko 𝑆))
34	18, 33	eqeltrd 2865	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} ∈ (𝑇 ↑ko 𝑆))
35		imaeq2 6049	. . . . . . . . 9 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) = (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}))
36		eqid 2765	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹))
37	36	mptpreima 6229	. . . . . . . . 9 ⊢ (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}}
38	35, 37	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}})
39	38	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → ((◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆) ↔ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}} ∈ (𝑇 ↑ko 𝑆)))
40	34, 39	syl5ibrcom 250	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆)))
41	40	expimpd 458	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) → (((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆)))
42	41	rexlimdvva 3222	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑇 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑥 = {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆)))
43	9, 42	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) → (◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆)))
44	43	ralrimiv 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})(◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆))
45		eqid 2765	. . . . 5 ⊢ (𝑇 ↑ko 𝑆) = (𝑇 ↑ko 𝑆)
46	45	xkotopon 23718	. . . 4 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
47	21, 23, 46	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48		ovex 7433	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
49	48	pwex 5342	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
50	5, 6, 7	xkotf 23703	. . . . . 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)
51		frn 6703	. . . . . 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))
52	50, 51	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)
53	49, 52	ssexi 5283	. . . 4 ⊢ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V
54	53	a1i 11	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣}) ∈ V)
55		cntop1 23358	. . . . 5 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
56	1, 55	syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Top)
57	5, 6, 7	xkoval 23705	. . . 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}))))
58	56, 23, 57	syl2anc 595	. . 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}))))
59		eqid 2765	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
60	59	xkotopon 23718	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
61	56, 23, 60	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
62	47, 54, 58, 61	subbascn 23372	. 2 ⊢ (𝜑 → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)) ↔ ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇) ∧ ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣})(◡(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) “ 𝑥) ∈ (𝑇 ↑ko 𝑆))))
63	4, 44, 62	mpbir2and 725	1 ⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868   ↦ cmpt 5186   × cxp 5650  ^◡ccnv 5651  ran crn 5653   “ cima 5655   ∘ ccom 5656  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ficfi 9358   ↾_t crest 17463  topGenctg 17480  Topctop 23011  TopOnctopon 23028   Cn ccn 23342  Compccmp 23504   ↑_ko cxko 23679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-1o 8441  df-2o 8442  df-map 8814  df-en 8932  df-dom 8933  df-fin 8935  df-fi 9359  df-rest 17465  df-topgen 17486  df-top 23012  df-topon 23029  df-bases 23064  df-cn 23345  df-cmp 23505  df-xko 23681

This theorem is referenced by:  cnmpt1k  23800