Theorem sssmf 44161

Description: The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
sssmf.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
sssmf.f	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))

Assertion

Ref	Expression
sssmf	⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Proof of Theorem sssmf

Dummy variables 𝑎 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1918	. 2 ⊢ Ⅎ𝑎𝜑
2		sssmf.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		inss2 4160	. . 3 ⊢ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4		sssmf.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
5		eqid 2738	. . . 4 ⊢ dom 𝐹 = dom 𝐹
6	2, 4, 5	smfdmss 44156	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆)
7	3, 6	sstrid 3928	. 2 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ ∪ 𝑆)
8	2, 4, 5	smff 44155	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
9	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10		fssres 6624	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
11	8, 9, 10	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
12	8	freld 6590	. . . . . . 7 ⊢ (𝜑 → Rel 𝐹)
13		resindm 5929	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵))
15	14	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16		dmres 5902	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹))
18	15, 17	feq12d 6572	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
19	11, 18	mpbird 256	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ)
20	17	feq2d 6570	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ 𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
21	19, 20	mpbid 231	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ)
22	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
23	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
25	22, 23, 5, 24	smfpreimalt 44154	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))
26	4	dmexd 7726	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ∈ V)
27		elrest 17055	. . . . . 6 ⊢ ((𝑆 ∈ SAlg ∧ dom 𝐹 ∈ V) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
28	2, 26, 27	syl2anc 583	. . . . 5 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
29	28	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹) ↔ ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
30	25, 29	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31		elinel1 4125	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ 𝐵)
32	31	fvresd 6776	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
33	32	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹 ↾ 𝐵)‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
34	33	rabbiia 3396	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
35	34	a1i 11	. . . . . . . . 9 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
36		rabss2 4007	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∩ dom 𝐹) ⊆ dom 𝐹 → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
37	3, 36	ax-mp 5	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
38		id 22	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
39		inss1 4159	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ dom 𝐹) ⊆ 𝑤
40	39	a1i 11	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) ⊆ 𝑤)
41	38, 40	eqsstrd 3955	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
42	37, 41	sstrid 3928	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
43		ssrab2 4009	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝐵 ∩ dom 𝐹)
44	43	a1i 11	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝐵 ∩ dom 𝐹))
45	42, 44	ssind 4163	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46		nfrab1 3310	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
47		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑤 ∩ dom 𝐹)
48	46, 47	nfeq 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49		elinel2 4126	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
50	49	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51		elinel1 4125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ 𝑤)
52	3	sseli 3913	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
53	49, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
54	51, 53	elind 4124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
55	54	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
56	38	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
58	55, 57	eleqtrd 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
59		rabid 3304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
60	59	biimpi 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
61	60	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} → (𝐹‘𝑥) < 𝑎)
62	58, 61	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝐹‘𝑥) < 𝑎)
63	50, 62	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
64		rabid 3304	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
65	63, 64	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
66	65	ex 412	. . . . . . . . . . . . . . 15 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}))
67	48, 66	ralrimi 3139	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
68		nfcv 2906	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69		nfrab1 3310	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
70	68, 69	dfss3f 3908	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ↔ ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
71	67, 70	sylibr 233	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
72	71	sseld 3916	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}))
73	48, 72	ralrimi 3139	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
74	73, 70	sylibr 233	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
75	45, 74	eqssd 3934	. . . . . . . . 9 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
76	35, 75	eqtrd 2778	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
77	76	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
78	77	3adant2 1129	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
79	22	3ad2ant1 1131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80		simp1l 1195	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
81	26, 9	ssexd 5243	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ∈ V)
82	80, 81	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝐵 ∩ dom 𝐹) ∈ V)
83		simp2 1135	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤 ∈ 𝑆)
84		eqid 2738	. . . . . . 7 ⊢ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
85	79, 82, 83, 84	elrestd 42547	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))
86	78, 85	eqeltrd 2839	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))
87	86	3exp 1117	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑤 ∈ 𝑆 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))))
88	87	rexlimdv 3211	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹))))
89	30, 88	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))
90	1, 2, 7, 21, 89	issmfd 44158	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  Rel wrel 5585  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℝcr 10801   < clt 10940   ↾_t crest 17048  SAlgcsalg 43739  SMblFncsmblfn 44123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-ioo 13012  df-ico 13014  df-rest 17050  df-smblfn 44124

This theorem is referenced by:  sssmfmpt  44173