Theorem sssmf 47090

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 1916	. 2 ⊢ Ⅎ𝑎𝜑
2		sssmf.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		inss2 4192	. . 3 ⊢ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4		sssmf.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
5		eqid 2737	. . . 4 ⊢ dom 𝐹 = dom 𝐹
6	2, 4, 5	smfdmss 47085	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆)
7	3, 6	sstrid 3947	. 2 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ ∪ 𝑆)
8	2, 4, 5	smff 47084	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
9	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10		fssres 6708	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
11	8, 9, 10	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
12	8	freld 6676	. . . . . . 7 ⊢ (𝜑 → Rel 𝐹)
13		resindm 5997	. . . . . . 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 2743	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16		dmres 5979	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹))
18	15, 17	feq12d 6658	. . . 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 6654	. . 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 47083	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))
26	4	dmexd 7855	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ∈ V)
27		elrest 17359	. . . . . 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 585	. . . . 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 4155	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ 𝐵)
32	31	fvresd 6862	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
33	32	breq1d 5110	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹 ↾ 𝐵)‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
34	33	rabbiia 3405	. . . . . . . . . 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 4191	. . . . . . . . . . . . . 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 3970	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
42	37, 41	sstrid 3947	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
43		ssrab2 4034	. . . . . . . . . . . 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 4195	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46		nfrab1 3421	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
47		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑤 ∩ dom 𝐹)
48	46, 47	nfeq 2913	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49		elinel2 4156	. . . . . . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ 𝑤)
52	3	sseli 3931	. . . . . . . . . . . . . . . . . . . . 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 4154	. . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
59		rabid 3422	. . . . . . . . . . . . . . . . . . 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 3422	. . . . . . . . . . . . . . 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 3236	. . . . . . . . . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69		nfrab1 3421	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
70	68, 69	dfss3f 3927	. . . . . . . . . . . 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 3953	. . . . . . . . 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 2772	. . . . . . . 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 1132	. . . . . 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 1134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
78		simp1l 1199	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
79	26, 9	ssexd 5271	. . . . . . . 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 1138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤 ∈ 𝑆)
82		eqid 2737	. . . . . . 7 ⊢ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
83	77, 80, 81, 82	elrestd 45461	. . . . . 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 2837	. . . . 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 1120	. . . 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 3137	. . 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 47087	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∪ cuni 4865   class class class wbr 5100  dom cdm 5632   ↾ cres 5634  Rel wrel 5637  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℝcr 11037   < clt 11178   ↾_t crest 17352  SAlgcsalg 46660  SMblFncsmblfn 47047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ioo 13277  df-ico 13279  df-rest 17354  df-smblfn 47048

This theorem is referenced by:  sssmfmpt  47102