Theorem liminfval2 45689

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 7455	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
4	3	imaeq2d 6089	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
5	4	ineq1d 4240	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
6	5	infeq1d 9546	. . . . . 6 ⊢ (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	6	cbvmptv 5279	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
8	2, 7	eqtri 2768	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
9	8	liminfval 45680	. . 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 45347	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
14		supxrunb1 13381	. . . . . . . 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 45679	. . . . . . . . . . 11 ⊢ 𝐺:ℝ⟶ℝ*
18	17	ffvelcdmi 7117	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ*)
19	18	ad2antlr 726	. . . . . . . . 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 4008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
23	17	ffvelcdmi 7117	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) ∈ ℝ*)
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ*)
25	20, 21, 24	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ ℝ*)
26		imassrn 6100	. . . . . . . . . . . 12 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
27		frn 6754	. . . . . . . . . . . . 13 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
28	17, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran 𝐺 ⊆ ℝ*
29	26, 28	sstri 4018	. . . . . . . . . . 11 ⊢ (𝐺 “ 𝐴) ⊆ ℝ*
30		supxrcl 13377	. . . . . . . . . . 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 583	. . . . . . . . . . 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 45675	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
37	33, 34, 35, 36	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
38	8	liminfgval 45683	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
39	38	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
40	8	liminfgval 45683	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
43	39, 42	breq12d 5179	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛) ≤ (𝐺‘𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
44	43	adantrr 716	. . . . . . . . . 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 1913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝜑
48		inss2 4259	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49		infxrcl 13395	. . . . . . . . . . . . . . . 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 6721	. . . . . . . . . . . . 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 7271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
56		supxrub 13386	. . . . . . . . . . 11 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
57	46, 55, 56	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
58	20, 21, 57	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
59	19, 25, 32, 45, 58	xrletrd 13224	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
60	59	rexlimdvaa 3162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
61	60	ralimdva 3173	. . . . . 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 13203	. . . . . . . . 9 ⊢ < Or ℝ*
64	63	infex 9562	. . . . . . . 8 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
65	64	rgenw 3071	. . . . . . 7 ⊢ ∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
66	8	fnmpt 6720	. . . . . . 7 ⊢ (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
67	65, 66	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
68		breq1 5169	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
69	68	ralrn 7122	. . . . . 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 13388	. . . . 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 691	. . . 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 13394	. . . 4 ⊢ (((𝐺 “ 𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
78	75, 76, 77	syl2anc 583	. . 3 ⊢ (𝜑 → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79		supxrcl 13377	. . . . 5 ⊢ (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
80	28, 79	ax-mp 5	. . . 4 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81		xrletri3 13216	. . . 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 691	. . 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 582	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ))
84	10, 83	eqtrd 2780	1 ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701   “ cima 5703   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  supcsup 9509  infcinf 9510  ℝcr 11183  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  [,)cico 13409  lim infclsi 45672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-ico 13413  df-liminf 45673

This theorem is referenced by:  liminfresico  45692