Theorem infnsuprnmpt 43599

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 2731	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infnsuprnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 43538	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		infnsuprnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6201	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		infnsuprnmpt.l	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
8	7	rnmptlb 43591	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
9		infrenegsup 12147	. . 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 1371	. 2 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
12		rabidim2 43434	. . . . . . . . . . . 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 11408	. . . . . . . . . . . 12 ⊢ -𝑤 ∈ V
15	2	elrnmpt 5916	. . . . . . . . . . . 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 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
18		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑤
19	18	nfneg 11406	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥-𝑤
20		nfmpt1 5218	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
21	20	nfrn 5912	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
22	19, 21	nfel 2916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
23		nfcv 2902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
24	22, 23	nfrabw 3441	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
25	18, 24	nfel 2916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
26	1, 25	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
27		rabidim1 3426	. . . . . . . . . . . . 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 11402	. . . . . . . . . . . . . . . 16 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
30	29	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31	30	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32		simp1r 1198	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33		recn 11150	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
34	33	negnegd 11512	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
35	32, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
36	31, 35	eqtr2d 2772	. . . . . . . . . . . . 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 3242	. . . . . . . . . 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 5921	. . . . . . . 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 3450	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
45	11	elrnmpt 5916	. . . . . . . . . . . . 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 215	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
48	47	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
49	18, 23	nfel 2916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
50	49, 22	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
51		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 = -𝐵)
52	3	renegcld 11591	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
53	52	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝐵 ∈ ℝ)
54	51, 53	eqeltrd 2832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑥 ∈ 𝐴)
56	51	negeqd 11404	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
57	3	recnd 11192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
58	57	negnegd 11512	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
59	58	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
60	56, 59	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
61		rspe 3230	. . . . . . . . . . . . . . . 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 5921	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
65	54, 64	jca 512	. . . . . . . . . . . . 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 3247	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))))
68	67	imp 407	. . . . . . . . . 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 3425	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
71	69, 70	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
72	71	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}))
73	43, 72	impbid 211	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
74	73	alrimiv 1930	. . . . 5 ⊢ (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
75		nfrab1 3424	. . . . . 6 ⊢ Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
76		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑤ran (𝑥 ∈ 𝐴 ↦ -𝐵)
77	75, 76	cleqf 2933	. . . . 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 233	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
79	78	supeq1d 9391	. . 3 ⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
80	79	negeqd 11404	. 2 ⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
81		eqidd 2732	. 2 ⊢ (𝜑 → -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
82	10, 80, 81	3eqtrd 2775	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3405  Vcvv 3446   ⊆ wss 3913  ∅c0 4287   class class class wbr 5110   ↦ cmpt 5193  ran crn 5639  supcsup 9385  infcinf 9386  ℝcr 11059   < clt 11198   ≤ cle 11199  -cneg 11395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9387  df-inf 9388  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397

This theorem is referenced by:  smfinflem  45178