Theorem infnsuprnmpt 45274

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 2735	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infnsuprnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 45220	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		infnsuprnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6233	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		infnsuprnmpt.l	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
8	7	rnmptlb 45267	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
9		infrenegsup 12225	. . 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 1373	. 2 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2735	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
12		rabidim2 45126	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13	12	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
14		negex 11480	. . . . . . . . . . . 12 ⊢ -𝑤 ∈ V
15	2	elrnmpt 5938	. . . . . . . . . . . 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 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
18		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑤
19	18	nfneg 11478	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥-𝑤
20		nfmpt1 5220	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
21	20	nfrn 5932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
22	19, 21	nfel 2913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
23		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
24	22, 23	nfrabw 3454	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
25	18, 24	nfel 2913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
26	1, 25	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
27		rabidim1 3438	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ℝ)
28	27	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ℝ)
29		negeq 11474	. . . . . . . . . . . . . . . 16 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
30	29	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31	30	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32		simp1r 1199	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33		recn 11219	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
34	33	negnegd 11585	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
35	32, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
36	31, 35	eqtr2d 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37	36	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
38	28, 37	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
39	26, 38	reximdai 3244	. . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
42	11, 40, 41	elrnmptd 5943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
43	42	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
44		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
45	11	elrnmpt 5938	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
46	44, 45	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
47	46	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
48	47	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
49	18, 23	nfel 2913	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
50	49, 22	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
51		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 = -𝐵)
52	3	renegcld 11664	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
53	52	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝐵 ∈ ℝ)
54	51, 53	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑥 ∈ 𝐴)
56	51	negeqd 11476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
57	3	recnd 11263	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
58	57	negnegd 11585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
59	58	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
60	56, 59	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
61		rspe 3232	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
62	55, 60, 61	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
63	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ V)
64	2, 62, 63	elrnmptd 5943	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
65	54, 64	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
66	65	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))))
67	1, 50, 66	rexlimd 3249	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))))
68	67	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
69	48, 68	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
70		rabid 3437	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
71	69, 70	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
72	71	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}))
73	43, 72	impbid 212	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
74	73	alrimiv 1927	. . . . 5 ⊢ (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
75		nfrab1 3436	. . . . . 6 ⊢ Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
76		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑤ran (𝑥 ∈ 𝐴 ↦ -𝐵)
77	75, 76	cleqf 2927	. . . . 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)))
78	74, 77	sylibr 234	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
79	78	supeq1d 9458	. . 3 ⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
80	79	negeqd 11476	. 2 ⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
81		eqidd 2736	. 2 ⊢ (𝜑 → -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
82	10, 80, 81	3eqtrd 2774	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119   ↦ cmpt 5201  ran crn 5655  supcsup 9452  infcinf 9453  ℝcr 11128   < clt 11269   ≤ cle 11270  -cneg 11467

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469

This theorem is referenced by:  smfinflem  46846