Theorem sssmf 46720

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 1914	. 2 ⊢ Ⅎ𝑎𝜑
2		sssmf.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		inss2 4191	. . 3 ⊢ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4		sssmf.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
5		eqid 2729	. . . 4 ⊢ dom 𝐹 = dom 𝐹
6	2, 4, 5	smfdmss 46715	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆)
7	3, 6	sstrid 3949	. 2 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ ∪ 𝑆)
8	2, 4, 5	smff 46714	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
9	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10		fssres 6694	. . . . 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 6662	. . . . . . 7 ⊢ (𝜑 → Rel 𝐹)
13		resindm 5985	. . . . . . 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 2735	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16		dmres 5967	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹))
18	15, 17	feq12d 6644	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
19	11, 18	mpbird 257	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ)
20	17	feq2d 6640	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ 𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
21	19, 20	mpbid 232	. 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 46713	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))
26	4	dmexd 7843	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ∈ V)
27		elrest 17349	. . . . . 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 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 232	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31		elinel1 4154	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ 𝐵)
32	31	fvresd 6846	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
33	32	breq1d 5105	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹 ↾ 𝐵)‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
34	33	rabbiia 3400	. . . . . . . . . 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 4031	. . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . 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 3972	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
42	37, 41	sstrid 3949	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
43		ssrab2 4033	. . . . . . . . . . . 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 4194	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46		nfrab1 3417	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
47		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑤 ∩ dom 𝐹)
48	46, 47	nfeq 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49		elinel2 4155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
50	49	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51		elinel1 4154	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ 𝑤)
52	3	sseli 3933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
53	49, 52	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
54	51, 53	elind 4153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
55	54	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
56	38	eqcomd 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
57	56	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (({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 2830	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
59		rabid 3418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
60	59	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
61	60	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} → (𝐹‘𝑥) < 𝑎)
62	58, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝐹‘𝑥) < 𝑎)
63	50, 62	jca 511	. . . . . . . . . . . . . . 15 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
64		rabid 3418	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
65	63, 64	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (({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	. . . . . . . . . . . . 13 ⊢ ({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 3227	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
68		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69		nfrab1 3417	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
70	68, 69	dfss3f 3929	. . . . . . . . . . . 12 ⊢ ((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 234	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
72	38, 38, 38, 71	4syl 19	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
73	45, 72	eqssd 3955	. . . . . . . . 9 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
74	35, 73	eqtrd 2764	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
75	74	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)))
76	75	3adant2 1131	. . . . . 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)))
77	22	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
78		simp1l 1198	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
79	26, 9	ssexd 5266	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ∈ V)
80	78, 79	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝐵 ∩ dom 𝐹) ∈ V)
81		simp2 1137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤 ∈ 𝑆)
82		eqid 2729	. . . . . . 7 ⊢ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
83	77, 80, 81, 82	elrestd 45086	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))
84	76, 83	eqeltrd 2828	. . . . 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)))
85	84	3exp 1119	. . . 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)))))
86	85	rexlimdv 3128	. . 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))))
87	30, 86	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹 ↾ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t (𝐵 ∩ dom 𝐹)))
88	1, 2, 7, 21, 87	issmfd 46717	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3396  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ∪ cuni 4861   class class class wbr 5095  dom cdm 5623   ↾ cres 5625  Rel wrel 5628  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  ℝcr 11027   < clt 11168   ↾_t crest 17342  SAlgcsalg 46290  SMblFncsmblfn 46677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-pre-lttri 11102  ax-pre-lttrn 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-er 8632  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-ioo 13270  df-ico 13272  df-rest 17344  df-smblfn 46678

This theorem is referenced by:  sssmfmpt  46732