Theorem liminfval2 45783

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 7438	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
4	3	imaeq2d 6078	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
5	4	ineq1d 4219	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
6	5	infeq1d 9517	. . . . . 6 ⊢ (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	6	cbvmptv 5255	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
8	2, 7	eqtri 2765	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
9	8	liminfval 45774	. . 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 45443	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
14		supxrunb1 13361	. . . . . . . 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 45773	. . . . . . . . . . 11 ⊢ 𝐺:ℝ⟶ℝ*
18	17	ffvelcdmi 7103	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ*)
19	18	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ*)
20		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝜑)
21		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
22	12	sselda 3983	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
23	17	ffvelcdmi 7103	. . . . . . . . . . 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 6089	. . . . . . . . . . . 12 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
27		frn 6743	. . . . . . . . . . . . 13 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
28	17, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran 𝐺 ⊆ ℝ*
29	26, 28	sstri 3993	. . . . . . . . . . 11 ⊢ (𝐺 “ 𝐴) ⊆ ℝ*
30		supxrcl 13357	. . . . . . . . . . 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 769	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
36		liminfgord 45769	. . . . . . . . . . 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 45777	. . . . . . . . . . . . 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 45777	. . . . . . . . . . . . . 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 5156	. . . . . . . . . . 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 4238	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49		infxrcl 13375	. . . . . . . . . . . . . . . 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 6709	. . . . . . . . . . . . 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 7254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
56		supxrub 13366	. . . . . . . . . . 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 13204	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
60	59	rexlimdvaa 3156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
61	60	ralimdva 3167	. . . . . 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 13183	. . . . . . . . 9 ⊢ < Or ℝ*
64	63	infex 9533	. . . . . . . 8 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
65	64	rgenw 3065	. . . . . . 7 ⊢ ∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
66	8	fnmpt 6708	. . . . . . 7 ⊢ (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
67	65, 66	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
68		breq1 5146	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
69	68	ralrn 7108	. . . . . 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 13368	. . . . 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 13374	. . . 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 13357	. . . . 5 ⊢ (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
80	28, 79	ax-mp 5	. . . 4 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81		xrletri3 13196	. . . 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 2777	1 ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686   “ cima 5688   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  supcsup 9480  infcinf 9481  ℝcr 11154  +∞cpnf 11292  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  [,)cico 13389  lim infclsi 45766

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-ico 13393  df-liminf 45767

This theorem is referenced by:  liminfresico  45786