Theorem sssmf 44274

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 1917	. 2 ⊢ Ⅎ𝑎𝜑
2		sssmf.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		inss2 4163	. . 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 44269	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆)
7	3, 6	sstrid 3932	. 2 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ ∪ 𝑆)
8	2, 4, 5	smff 44268	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
9	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10		fssres 6640	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
12	8	freld 6606	. . . . . . 7 ⊢ (𝜑 → Rel 𝐹)
13		resindm 5940	. . . . . . 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 5913	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹))
18	15, 17	feq12d 6588	. . . 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 6586	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
23	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
25	22, 23, 5, 24	smfpreimalt 44267	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))
26	4	dmexd 7752	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ∈ V)
27		elrest 17138	. . . . . 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 584	. . . . 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 481	. . . 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 4129	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ 𝐵)
32	31	fvresd 6794	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
33	32	breq1d 5084	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹 ↾ 𝐵)‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
34	33	rabbiia 3407	. . . . . . . . . 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 4011	. . . . . . . . . . . . 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 4162	. . . . . . . . . . . . . 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 3959	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
42	37, 41	sstrid 3932	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
43		ssrab2 4013	. . . . . . . . . . . 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 4166	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46		nfrab1 3317	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
47		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑤 ∩ dom 𝐹)
48	46, 47	nfeq 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49		elinel2 4130	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
50	49	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51		elinel1 4129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ 𝑤)
52	3	sseli 3917	. . . . . . . . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
55	54	adantl 482	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 3310	. . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
64		rabid 3310	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . 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 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69		nfrab1 3317	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
70	68, 69	dfss3f 3912	. . . . . . . . . . . . . 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 3920	. . . . . . . . . . . 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 3141	. . . . . . . . . . 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 3938	. . . . . . . . 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 482	. . . . . . 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 1130	. . . . . 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 1132	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80		simp1l 1196	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
81	26, 9	ssexd 5248	. . . . . . . 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 1136	. . . . . . 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 42658	. . . . . 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 1118	. . . 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 3212	. . 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 44271	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  Rel wrel 5594  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℝcr 10870   < clt 11009   ↾_t crest 17131  SAlgcsalg 43849  SMblFncsmblfn 44233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ioo 13083  df-ico 13085  df-rest 17133  df-smblfn 44234

This theorem is referenced by:  sssmfmpt  44286