Theorem xkoco1cn 23631

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 23632 independently of the more general xkococn 23634 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 23240	. . . 4 ⊢ ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
3	1, 2	sylan 581	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
4	3	fmpttd 7059	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
5		eqid 2737	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2737	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑦) ∈ Comp}
7		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})
8	5, 6, 7	xkobval 23560	. . . . 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 2879	. . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
11	10, 2	sylan 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
12		imaeq1 6012	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑔 ∘ 𝐹) → (ℎ “ 𝑘) = ((𝑔 ∘ 𝐹) “ 𝑘))
13		imaco 6207	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∘ 𝐹) “ 𝑘) = (𝑔 “ (𝐹 “ 𝑘))
14	12, 13	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑔 ∘ 𝐹) → (ℎ “ 𝑘) = (𝑔 “ (𝐹 “ 𝑘)))
15	14	sseq1d 3954	. . . . . . . . . . 11 ⊢ (ℎ = (𝑔 ∘ 𝐹) → ((ℎ “ 𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣))
16	15	elrab3 3636	. . . . . . . . . 10 ⊢ ((𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇) → ((𝑔 ∘ 𝐹) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹 “ 𝑘)) ⊆ 𝑣))
17	11, 16	syl 17	. . . . . . . . 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 3396	. . . . . . . 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 2737	. . . . . . . . 9 ⊢ ∪ 𝑆 = ∪ 𝑆
20		cntop2 23215	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
21	1, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Top)
22	21	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
23		xkoco1cn.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ Top)
24	23	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
25		imassrn 6028	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑘) ⊆ ran 𝐹
26	5, 19	cnf 23220	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹:∪ 𝑅⟶∪ 𝑆)
27		frn 6667	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝑅⟶∪ 𝑆 → ran 𝐹 ⊆ ∪ 𝑆)
28	10, 26, 27	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → ran 𝐹 ⊆ ∪ 𝑆)
29	25, 28	sstrid 3934	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝐹 “ 𝑘) ⊆ ∪ 𝑆)
30		imacmp 23371	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑆 ↾t (𝐹 “ 𝑘)) ∈ Comp)
31	10, 30	sylancom 589	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → (𝑆 ↾t (𝐹 “ 𝑘)) ∈ Comp)
32		simplrr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑇)
33	19, 22, 24, 29, 31, 32	xkoopn 23563	. . . . . . . 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 2837	. . . . . . 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 6013	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹))
37	36	mptpreima 6194	. . . . . . . . 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 2788	. . . . . . . 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 2822	. . . . . . 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 247	. . . . . 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 453	. . . . 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 3195	. . . 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 242	. . 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 3129	. 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 2737	. . . . 5 ⊢ (𝑇 ↑ko 𝑆) = (𝑇 ↑ko 𝑆)
46	45	xkotopon 23574	. . . 4 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
47	21, 23, 46	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48		ovex 7391	. . . . . 6 ⊢ (𝑅 Cn 𝑇) ∈ V
49	48	pwex 5315	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑇) ∈ V
50	5, 6, 7	xkotf 23559	. . . . . 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 6667	. . . . . 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 5257	. . . 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 23214	. . . . 5 ⊢ (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
56	1, 55	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Top)
57	5, 6, 7	xkoval 23561	. . . 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 585	. . 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 2737	. . . . 5 ⊢ (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
60	59	xkotopon 23574	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
61	56, 23, 60	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑇 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
62	47, 54, 58, 61	subbascn 23228	. 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 714	1 ⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   ↦ cmpt 5167   × cxp 5620  ^◡ccnv 5621  ran crn 5623   “ cima 5625   ∘ ccom 5626  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360  ficfi 9314   ↾_t crest 17372  topGenctg 17389  Topctop 22867  TopOnctopon 22884   Cn ccn 23198  Compccmp 23360   ↑_ko cxko 23535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-1o 8396  df-2o 8397  df-map 8766  df-en 8885  df-dom 8886  df-fin 8888  df-fi 9315  df-rest 17374  df-topgen 17395  df-top 22868  df-topon 22885  df-bases 22920  df-cn 23201  df-cmp 23361  df-xko 23537

This theorem is referenced by:  cnmpt1k  23656