Theorem infnsuprnmpt 45701

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 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infnsuprnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 7072	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		infnsuprnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6204	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		infnsuprnmpt.l	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
8	7	rnmptlb 45694	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
9		infrenegsup 12134	. . 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 1374	. 2 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
12		rabidim2 45554	. . . . . . . . . . . 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 11386	. . . . . . . . . . . 12 ⊢ -𝑤 ∈ V
15	2	elrnmpt 5909	. . . . . . . . . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑤
19	18	nfneg 11384	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥-𝑤
20		nfmpt1 5185	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
21	20	nfrn 5903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
22	19, 21	nfel 2914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
23		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
24	22, 23	nfrabw 3427	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
25	18, 24	nfel 2914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
26	1, 25	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
27		rabidim1 3412	. . . . . . . . . . . . 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 11380	. . . . . . . . . . . . . . . 16 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
30	29	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31	30	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32		simp1r 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33		recn 11123	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
34	33	negnegd 11491	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
35	32, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
36	31, 35	eqtr2d 2773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37	36	3exp 1120	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
38	28, 37	syldan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵)))
39	26, 38	reximdai 3240	. . . . . . . . . 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 5914	. . . . . . . 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 3434	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
45	11	elrnmpt 5909	. . . . . . . . . . . . 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 2914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
50	49, 22	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
51		simp3 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 = -𝐵)
52	3	renegcld 11572	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
53	52	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝐵 ∈ ℝ)
54	51, 53	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55		simp2 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑥 ∈ 𝐴)
56	51	negeqd 11382	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
57	3	recnd 11168	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
58	57	negnegd 11491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
59	58	3adant3 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
60	56, 59	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
61		rspe 3228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
62	55, 60, 61	syl2anc 585	. . . . . . . . . . . . . . 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 5914	. . . . . . . . . . . . . 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 1120	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))))
67	1, 50, 66	rexlimd 3245	. . . . . . . . . . 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 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
70		rabid 3411	. . . . . . . . 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 1929	. . . . 5 ⊢ (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
75		nfrab1 3410	. . . . . 6 ⊢ Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
76		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑤ran (𝑥 ∈ 𝐴 ↦ -𝐵)
77	75, 76	cleqf 2928	. . . . 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 9354	. . 3 ⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
80	79	negeqd 11382	. 2 ⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
81		eqidd 2738	. 2 ⊢ (𝜑 → -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
82	10, 80, 81	3eqtrd 2776	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086   ↦ cmpt 5167  ran crn 5627  supcsup 9348  infcinf 9349  ℝcr 11032   < clt 11174   ≤ cle 11175  -cneg 11373

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-po 5534  df-so 5535  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-sup 9350  df-inf 9351  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375

This theorem is referenced by:  smfinflem  47267