Theorem liminfval2 45739

Description: The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfval2.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
liminfval2.2	⊢ (𝜑 → 𝐹 ∈ 𝑉)
liminfval2.3	⊢ (𝜑 → 𝐴 ⊆ ℝ)
liminfval2.4	⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)

Assertion

Ref	Expression
liminfval2	⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Distinct variable group:   𝑘,𝐹

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval2

Dummy variables 𝑛 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		liminfval2.2	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		liminfval2.1	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
3		oveq1 7376	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
4	3	imaeq2d 6020	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
5	4	ineq1d 4178	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
6	5	infeq1d 9405	. . . . . 6 ⊢ (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	6	cbvmptv 5206	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
8	2, 7	eqtri 2752	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
9	8	liminfval 45730	. . 3 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
10	1, 9	syl 17	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
11		liminfval2.4	. . . . . . 7 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
12		liminfval2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
13	12	ssrexr 45401	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
14		supxrunb1 13255	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
16	11, 15	mpbird 257	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥)
17	8	liminfgf 45729	. . . . . . . . . . 11 ⊢ 𝐺:ℝ⟶ℝ*
18	17	ffvelcdmi 7037	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ*)
19	18	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ*)
20		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝜑)
21		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
22	12	sselda 3943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
23	17	ffvelcdmi 7037	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) ∈ ℝ*)
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ*)
25	20, 21, 24	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ ℝ*)
26		imassrn 6031	. . . . . . . . . . . 12 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
27		frn 6677	. . . . . . . . . . . . 13 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
28	17, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran 𝐺 ⊆ ℝ*
29	26, 28	sstri 3953	. . . . . . . . . . 11 ⊢ (𝐺 “ 𝐴) ⊆ ℝ*
30		supxrcl 13251	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐴) ⊆ ℝ* → sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*)
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*
32	31	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*)
33		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
34	20, 21, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
36		liminfgord 45725	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
37	33, 34, 35, 36	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
38	8	liminfgval 45733	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
39	38	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
40	8	liminfgval 45733	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
41	22, 40	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
42	41	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
43	39, 42	breq12d 5115	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛) ≤ (𝐺‘𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
44	43	adantrr 717	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → ((𝐺‘𝑛) ≤ (𝐺‘𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
45	37, 44	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ (𝐺‘𝑥))
46	29	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺 “ 𝐴) ⊆ ℝ*)
47		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝜑
48		inss2 4197	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49		infxrcl 13270	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ* → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
52	47, 51, 8	fnmptd 6641	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn ℝ)
53	52	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn ℝ)
54		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
55	53, 22, 54	fnfvimad 7190	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
56		supxrub 13260	. . . . . . . . . . 11 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
57	46, 55, 56	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
58	20, 21, 57	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
59	19, 25, 32, 45, 58	xrletrd 13098	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
60	59	rexlimdvaa 3135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
61	60	ralimdva 3145	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
62	16, 61	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
63		xrltso 13077	. . . . . . . . 9 ⊢ < Or ℝ*
64	63	infex 9422	. . . . . . . 8 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
65	64	rgenw 3048	. . . . . . 7 ⊢ ∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
66	8	fnmpt 6640	. . . . . . 7 ⊢ (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
67	65, 66	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
68		breq1 5105	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
69	68	ralrn 7042	. . . . . 6 ⊢ (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
70	67, 69	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
71	62, 70	sylibr 234	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
72		supxrleub 13262	. . . . 5 ⊢ ((ran 𝐺 ⊆ ℝ* ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
73	28, 31, 72	mp2an 692	. . . 4 ⊢ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
74	71, 73	sylibr 234	. . 3 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
75	26	a1i 11	. . . 4 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ ran 𝐺)
76	28	a1i 11	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ*)
77		supxrss 13268	. . . 4 ⊢ (((𝐺 “ 𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
78	75, 76, 77	syl2anc 584	. . 3 ⊢ (𝜑 → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79		supxrcl 13251	. . . . 5 ⊢ (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
80	28, 79	ax-mp 5	. . . 4 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81		xrletri3 13090	. . . 4 ⊢ ((sup(ran 𝐺, ℝ*, < ) ∈ ℝ* ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))))
82	80, 31, 81	mp2an 692	. . 3 ⊢ (sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < )))
83	74, 78, 82	sylanbrc 583	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ))
84	10, 83	eqtrd 2764	1 ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911   class class class wbr 5102   ↦ cmpt 5183  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  supcsup 9367  infcinf 9368  ℝcr 11043  +∞cpnf 11181  ℝ^*cxr 11183   < clt 11184   ≤ cle 11185  [,)cico 13284  lim infclsi 45722

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-ico 13288  df-liminf 45723

This theorem is referenced by:  liminfresico  45742