Theorem issmfle 40954

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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfle.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 40942	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 40941	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑏𝜑
8		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝜑
11		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
12	10, 11	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
13		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
14	12, 13	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
15		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
16	1	uniexd 39281	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
18		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
19	17, 18	ssexd 4805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
20	5, 19	syldan 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
21		eqid 2622	. . . . . . . . . 10 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
22	2, 20, 21	subsalsal 40577	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
23	22	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
24	6	frexr 39604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*)
25	24	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*)
26	25	ffvelrnda 6359	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ*)
27	2	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
28	3	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
29		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
30	27, 28, 4, 29	smfpreimalt 40940	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
31	30	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷))
32		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
33	14, 15, 23, 26, 31, 32	salpreimaltle 40935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
34	33	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
35	9, 34	ralrimi 2957	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
36	5, 6, 35	3jca 1242	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
37	36	ex 450	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
38		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
39		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
40		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
41		nfrab1 3122	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏}
42		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
43	41, 42	nfel 2777	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
44	40, 43	nfral 2945	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
45	38, 39, 44	nf3an 1831	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
46	10, 45	nfan 1828	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
47		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
48		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
49		nfra1 2941	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)
50	47, 48, 49	nf3an 1831	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
51	7, 50	nfan 1828	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))
52	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
53		simpr1 1067	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
54		simpr2 1068	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
55		rspa 2930	. . . . . . 7 ⊢ ((∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))
56	55	3ad2antl3 1225	. . . . . 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 750	. . . . 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 40953	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
59	58	ex 450	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
60	37, 59	impbid 202	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))))
61		breq2 4657	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → ((𝐹‘𝑦) ≤ 𝑏 ↔ (𝐹‘𝑦) ≤ 𝑎))
62	61	rabbidv 3189	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎})
63		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
64	63	breq1d 4663	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ≤ 𝑎 ↔ (𝐹‘𝑥) ≤ 𝑎))
65	64	cbvrabv 3199	. . . . . . . 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 2656	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎})
68	67	eleq1d 2686	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))
69	68	cbvralv 3171	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))
70	69	3anbi3i 1255	. . 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 268	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   ↾_t crest 16081  SAlgcsalg 40528  SMblFncsmblfn 40909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smfpreimale  40963  issmfgt  40965  issmfled  40966