Theorem sssmf 43005

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 1909	. 2 ⊢ Ⅎ𝑎𝜑
2		sssmf.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		inss2 4204	. . 3 ⊢ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4		sssmf.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
5		eqid 2819	. . . 4 ⊢ dom 𝐹 = dom 𝐹
6	2, 4, 5	smfdmss 43000	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆)
7	3, 6	sstrid 3976	. 2 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ ∪ 𝑆)
8	2, 4, 5	smff 42999	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
9	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10		fssres 6537	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
11	8, 9, 10	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
12	8	freld 41482	. . . . . . 7 ⊢ (𝜑 → Rel 𝐹)
13		resindm 5893	. . . . . . 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 2825	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16		dmres 5868	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹))
18	15, 17	feq12d 6495	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
19	11, 18	mpbird 259	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ)
20	17	feq2d 6493	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵):dom (𝐹 ↾ 𝐵)⟶ℝ ↔ (𝐹 ↾ 𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
21	19, 20	mpbid 234	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ)
22	2	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
23	4	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
25	22, 23, 5, 24	smfpreimalt 42998	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹))
26	4	dmexd 7607	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ∈ V)
27		elrest 16693	. . . . . 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 586	. . . . 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 483	. . . 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 234	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑤 ∈ 𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31		elinel1 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ 𝐵)
32	31	fvresd 6683	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
33	32	breq1d 5067	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹 ↾ 𝐵)‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
34	33	rabbiia 3471	. . . . . . . . . 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 4052	. . . . . . . . . . . . 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 4203	. . . . . . . . . . . . . 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 4003	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
42	37, 41	sstrid 3976	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ 𝑤)
43		ssrab2 4054	. . . . . . . . . . . 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 4207	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46		nfrab1 3383	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}
47		nfcv 2975	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑤 ∩ dom 𝐹)
48	46, 47	nfeq 2989	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49		elinel2 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
50	49	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51		elinel1 4170	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ 𝑤)
52	3	sseli 3961	. . . . . . . . . . . . . . . . . . . . . . 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 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
55	54	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
56	38	eqcomd 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
57	56	adantr 483	. . . . . . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎})
59		rabid 3377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
60	59	biimpi 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) < 𝑎))
61	60	simprd 498	. . . . . . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
64		rabid 3377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹‘𝑥) < 𝑎))
65	63, 64	sylibr 236	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 3214	. . . . . . . . . . . . . 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 2975	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69		nfrab1 3383	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}
70	68, 69	dfss3f 3957	. . . . . . . . . . . . . 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 236	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎})
72	71	sseld 3964	. . . . . . . . . . . 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 3214	. . . . . . . . . . 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 236	. . . . . . . . . 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 3982	. . . . . . . . 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 2854	. . . . . . . 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 484	. . . . . . 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 1126	. . . . . 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 1128	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80		simp1l 1192	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
81	26, 9	ssexd 5219	. . . . . . . 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 1132	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑤 ∈ 𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤 ∈ 𝑆)
84		eqid 2819	. . . . . . 7 ⊢ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
85	79, 82, 83, 84	elrestd 41364	. . . . . 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 2911	. . . . 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 1114	. . . 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 3281	. . 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 43002	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ∃wrex 3137  {crab 3140  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ∪ cuni 4830   class class class wbr 5057  dom cdm 5548   ↾ cres 5550  Rel wrel 5553  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  ℝcr 10528   < clt 10667   ↾_t crest 16686  SAlgcsalg 42583  SMblFncsmblfn 42967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-pre-lttri 10603  ax-pre-lttrn 10604

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-er 8281  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-ioo 12734  df-ico 12736  df-rest 16688  df-smblfn 42968

This theorem is referenced by:  sssmfmpt  43017