Theorem xkoopn 23499

Description: A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
xkoopn.x	⊢ 𝑋 = ∪ 𝑅
xkoopn.r	⊢ (𝜑 → 𝑅 ∈ Top)
xkoopn.s	⊢ (𝜑 → 𝑆 ∈ Top)
xkoopn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
xkoopn.c	⊢ (𝜑 → (𝑅 ↾t 𝐴) ∈ Comp)
xkoopn.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Assertion

Ref	Expression
xkoopn	⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ (𝑆 ↑ko 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   𝑈,𝑓

Allowed substitution hints:   𝜑(𝑓)   𝑋(𝑓)

Proof of Theorem xkoopn

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

Step	Hyp	Ref	Expression
1		ovex 7374	. . . . . . 7 ⊢ (𝑅 Cn 𝑆) ∈ V
2	1	pwex 5313	. . . . . 6 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
3		xkoopn.x	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝑅
4		eqid 2731	. . . . . . . 8 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
5		eqid 2731	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
6	3, 4, 5	xkotf 23495	. . . . . . 7 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
7		frn 6653	. . . . . . 7 ⊢ ((𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
9	2, 8	ssexi 5255	. . . . 5 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
10		ssfii 9298	. . . . 5 ⊢ (ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
11	9, 10	ax-mp 5	. . . 4 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
12		fvex 6830	. . . . 5 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V
13		bastg 22876	. . . . 5 ⊢ ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ V → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
14	12, 13	ax-mp 5	. . . 4 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
15	11, 14	sstri 3939	. . 3 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))
16		oveq2 7349	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑅 ↾t 𝑥) = (𝑅 ↾t 𝐴))
17	16	eleq1d 2816	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑅 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝐴) ∈ Comp))
18		xkoopn.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
19		xkoopn.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Top)
20	3	topopn 22816	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
21		elpw2g 5266	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑅 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
22	19, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
23	18, 22	mpbird 257	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
24		xkoopn.c	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾t 𝐴) ∈ Comp)
25	17, 23, 24	elrabd 3644	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp})
26		xkoopn.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
27		eqidd 2732	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈})
28		imaeq2 6000	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑓 “ 𝑘) = (𝑓 “ 𝐴))
29	28	sseq1d 3961	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ (𝑓 “ 𝐴) ⊆ 𝑣))
30	29	rabbidv 3402	. . . . . . 7 ⊢ (𝑘 = 𝐴 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑣})
31	30	eqeq2d 2742	. . . . . 6 ⊢ (𝑘 = 𝐴 → ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑣}))
32		sseq2 3956	. . . . . . . 8 ⊢ (𝑣 = 𝑈 → ((𝑓 “ 𝐴) ⊆ 𝑣 ↔ (𝑓 “ 𝐴) ⊆ 𝑈))
33	32	rabbidv 3402	. . . . . . 7 ⊢ (𝑣 = 𝑈 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈})
34	33	eqeq2d 2742	. . . . . 6 ⊢ (𝑣 = 𝑈 → ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈}))
35	31, 34	rspc2ev 3585	. . . . 5 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} ∧ 𝑈 ∈ 𝑆 ∧ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈}) → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
36	25, 26, 27, 35	syl3anc 1373	. . . 4 ⊢ (𝜑 → ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
37	1	rabex 5272	. . . . 5 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ V
38		eqeq1 2735	. . . . . 6 ⊢ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} → (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
39	38	2rexbidv 3197	. . . . 5 ⊢ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} → (∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
40	5	rnmpo 7474	. . . . 5 ⊢ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑦 ∣ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
41	37, 39, 40	elab2 3633	. . . 4 ⊢ ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
42	36, 41	sylibr 234	. . 3 ⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
43	15, 42	sselid 3927	. 2 ⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
44		xkoopn.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ Top)
45	3, 4, 5	xkoval 23497	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
46	19, 44, 45	syl2anc 584	. 2 ⊢ (𝜑 → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
47	43, 46	eleqtrrd 2834	1 ⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ (𝑆 ↑ko 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854   × cxp 5609  ran crn 5612   “ cima 5614  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  ficfi 9289   ↾_t crest 17319  topGenctg 17336  Topctop 22803   Cn ccn 23134  Compccmp 23296   ↑_ko cxko 23471

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-1o 8380  df-en 8865  df-fin 8868  df-fi 9290  df-topgen 17342  df-top 22804  df-xko 23473

This theorem is referenced by:  xkouni  23509  xkohaus  23563  xkoptsub  23564  xkoco1cn  23567  xkoco2cn  23568  xkococnlem  23569