Theorem issmfle 47279

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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfle.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 47267	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 47266	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1918	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10		nfv 1933	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝜑
11		nfv 1933	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
12	10, 11	nfan 1918	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
13		nfv 1933	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
14	12, 13	nfan 1918	. . . . . . . 8 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15		nfv 1933	. . . . . . . 8 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
16	1	uniexd 7719	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
17	16	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
18		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
19	17, 18	ssexd 5277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
20	5, 19	syldan 600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21		eqid 2761	. . . . . . . . . 10 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
22	2, 20, 21	subsalsal 46893	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
23	22	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
24	6	frexr 45920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
25	24	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
26	25	ffvelcdmda 7059	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ*)
27	2	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
28	3	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
30	27, 28, 4, 29	smfpreimalt 47265	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
31	30	adantlr 725	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
32		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
33	14, 15, 23, 26, 31, 32	salpreimaltle 47260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
34	33	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
35	9, 34	ralrimi 3259	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
36	5, 6, 35	3jca 1140	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
37	36	ex 416	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
38		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
39		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
40		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
41		nfrab1 3433	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏}
42		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
43	41, 42	nfel 2937	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
44	40, 43	nfralw 3308	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
45	38, 39, 44	nf3an 1920	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
46	10, 45	nfan 1918	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
47		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
48		nfv 1933	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
49		nfra1 3285	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
50	47, 48, 49	nf3an 1920	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
51	7, 50	nfan 1918	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
52	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
53		simpr1 1207	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
54		simpr2 1208	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
55		rspa 3250	. . . . . . 7 ⊢ ((∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
56	55	3ad2antl3 1200	. . . . . 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 724	. . . . 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 47278	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
59	58	ex 416	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
60	37, 59	impbid 214	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
61		breq2 5101	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → ((𝐹‘𝑦) ≤ 𝑏 ↔ (𝐹‘𝑦) ≤ 𝑎))
62	61	rabbidv 3420	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎})
63		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
64	63	breq1d 5107	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ≤ 𝑎 ↔ (𝐹‘𝑥) ≤ 𝑎))
65	64	cbvrabv 3423	. . . . . . . 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 2796	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎})
68	67	eleq1d 2846	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))
69	68	cbvralvw 3239	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))
70	69	3anbi3i 1171	. . 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 281	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4862   class class class wbr 5097  dom cdm 5643  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  ℝcr 11065  ℝ^*cxr 11208   < clt 11209   ≤ cle 11210   ↾_t crest 17439  SAlgcsalg 46842  SMblFncsmblfn 47229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-inf2 9589  ax-cc 10385  ax-ac2 10413  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9381  df-inf 9382  df-card 9890  df-acn 9893  df-ac 10065  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-n0 12475  df-z 12562  df-uz 12833  df-q 12943  df-rp 12987  df-ioo 13346  df-ico 13348  df-fl 13795  df-rest 17441  df-salg 46843  df-smblfn 47230

This theorem is referenced by:  smfpreimale  47288  issmfgt  47290  issmfled  47291