Theorem issmfle 46701

Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmfle.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmfle.d	⊢ 𝐷 = dom 𝐹

Assertion

Ref	Expression
issmfle	⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))))

Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎

Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfle

Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmfle.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfle.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 46689	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 46688	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1897	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10		nfv 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝜑
11		nfv 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
12	10, 11	nfan 1897	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
13		nfv 1912	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
14	12, 13	nfan 1897	. . . . . . . 8 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15		nfv 1912	. . . . . . . 8 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
16	1	uniexd 7761	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
18		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
19	17, 18	ssexd 5330	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
20	5, 19	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21		eqid 2735	. . . . . . . . . 10 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
22	2, 20, 21	subsalsal 46315	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
23	22	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
24	6	frexr 45335	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
25	24	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
26	25	ffvelcdmda 7104	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ*)
27	2	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
28	3	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
30	27, 28, 4, 29	smfpreimalt 46687	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
31	30	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
32		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
33	14, 15, 23, 26, 31, 32	salpreimaltle 46682	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
34	33	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
35	9, 34	ralrimi 3255	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
36	5, 6, 35	3jca 1127	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
37	36	ex 412	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
38		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
39		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
40		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
41		nfrab1 3454	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏}
42		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
43	41, 42	nfel 2918	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
44	40, 43	nfralw 3309	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
45	38, 39, 44	nf3an 1899	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
46	10, 45	nfan 1897	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
47		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
48		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
49		nfra1 3282	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
50	47, 48, 49	nf3an 1899	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
51	7, 50	nfan 1897	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
52	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
53		simpr1 1193	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
54		simpr2 1194	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
55		rspa 3246	. . . . . . 7 ⊢ ((∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
56	55	3ad2antl3 1186	. . . . . 6 ⊢ (((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
57	56	adantll 714	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
58	46, 51, 52, 4, 53, 54, 57	issmflelem 46700	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
59	58	ex 412	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
60	37, 59	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
61		breq2 5152	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → ((𝐹‘𝑦) ≤ 𝑏 ↔ (𝐹‘𝑦) ≤ 𝑎))
62	61	rabbidv 3441	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎})
63		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
64	63	breq1d 5158	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ≤ 𝑎 ↔ (𝐹‘𝑥) ≤ 𝑎))
65	64	cbvrabv 3444	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}
66	65	a1i 11	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎})
67	62, 66	eqtrd 2775	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎})
68	67	eleq1d 2824	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))
69	68	cbvralvw 3235	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))
70	69	3anbi3i 1158	. . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))
71	70	a1i 11	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))))
72	60, 71	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ⊆ wss 3963  ∪ cuni 4912   class class class wbr 5148  dom cdm 5689  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   ↾_t crest 17467  SAlgcsalg 46264  SMblFncsmblfn 46651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-ioo 13388  df-ico 13390  df-fl 13829  df-rest 17469  df-salg 46265  df-smblfn 46652

This theorem is referenced by:  smfpreimale  46710  issmfgt  46712  issmfled  46713