Theorem infnsuprnmpt 45702

Description: The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
infnsuprnmpt.x	⊢ Ⅎ𝑥𝜑
infnsuprnmpt.a	⊢ (𝜑 → 𝐴 ≠ ∅)
infnsuprnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
infnsuprnmpt.l	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)

Assertion

Ref	Expression
infnsuprnmpt	⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem infnsuprnmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infnsuprnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2739	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infnsuprnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 7066	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		infnsuprnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6196	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		infnsuprnmpt.l	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
8	7	rnmptlb 45695	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
9		infrenegsup 12131	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
10	4, 6, 8, 9	syl3anc 1379	. 2 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
12		rabidim2 45557	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13	12	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
14		negex 11383	. . . . . . . . . . . 12 ⊢ -𝑤 ∈ V
15	2	elrnmpt 5901	. . . . . . . . . . . 12 ⊢ (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
17	13, 16	sylib 219	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
18		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑤
19	18	nfneg 11381	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥-𝑤
20		nfmpt1 5172	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
21	20	nfrn 5895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
22	19, 21	nfel 2915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
23		nfcv 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
24	22, 23	nfrabw 3428	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
25	18, 24	nfel 2915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
26	1, 25	nfan 1906	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
27		rabidim1 3413	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ℝ)
28	27	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ℝ)
29		negeq 11377	. . . . . . . . . . . . . . . 16 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
30	29	eqcomd 2745	. . . . . . . . . . . . . . 15 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31	30	3ad2ant3 1141	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32		simp1r 1205	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33		recn 11120	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
34	33	negnegd 11488	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
35	32, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
36	31, 35	eqtr2d 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37	36	3exp 1125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
38	28, 37	syldan 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
39	26, 38	reximdai 3241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
40	17, 39	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
41		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
42	11, 40, 41	elrnmptd 5906	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
43	42	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
44		vex 3435	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
45	11	elrnmpt 5901	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
46	44, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
47	46	bilani 505	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
48	18, 23	nfel 2915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
49	48, 22	nfan 1906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
50		simp3 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 = -𝐵)
51	3	renegcld 11569	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
52	51	3adant3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝐵 ∈ ℝ)
53	50, 52	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 ∈ ℝ)
54		simp2 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑥 ∈ 𝐴)
55	50	negeqd 11379	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
56	3	recnd 11165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
57	56	negnegd 11488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
58	57	3adant3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
59	55, 58	eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
60		rspe 3229	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
61	54, 59, 60	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
62	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ V)
63	2, 61, 62	elrnmptd 5906	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
64	53, 63	jca 516	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
65	64	3exp 1125	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))))
66	1, 49, 65	rexlimd 3246	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))))
67	66	imp 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
68	47, 67	syldan 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
69		rabid 3412	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
70	68, 69	sylibr 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
71	70	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}))
72	43, 71	impbid 213	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
73	72	alrimiv 1934	. . . . 5 ⊢ (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
74		nfrab1 3411	. . . . . 6 ⊢ Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
75		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑤ran (𝑥 ∈ 𝐴 ↦ -𝐵)
76	74, 75	cleqf 2929	. . . . 5 ⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
77	73, 76	sylibr 235	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
78	77	supeq1d 9350	. . 3 ⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
79	78	negeqd 11379	. 2 ⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
80		eqidd 2740	. 2 ⊢ (𝜑 → -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
81	10, 79, 80	3eqtrd 2778	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ⊆ wss 3883  ∅c0 4262   class class class wbr 5073   ↦ cmpt 5154  ran crn 5620  supcsup 9344  infcinf 9345  ℝcr 11029   < clt 11171   ≤ cle 11172  -cneg 11370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9346  df-inf 9347  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372

This theorem is referenced by:  smfinflem  47268