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Theorem sssmf 43946
Description: The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
sssmf.s (𝜑𝑆 ∈ SAlg)
sssmf.f (𝜑𝐹 ∈ (SMblFn‘𝑆))
Assertion
Ref Expression
sssmf (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))

Proof of Theorem sssmf
Dummy variables 𝑎 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1922 . 2 𝑎𝜑
2 sssmf.s . 2 (𝜑𝑆 ∈ SAlg)
3 inss2 4144 . . 3 (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4 sssmf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
5 eqid 2737 . . . 4 dom 𝐹 = dom 𝐹
62, 4, 5smfdmss 43941 . . 3 (𝜑 → dom 𝐹 𝑆)
73, 6sstrid 3912 . 2 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ 𝑆)
82, 4, 5smff 43940 . . . . 5 (𝜑𝐹:dom 𝐹⟶ℝ)
93a1i 11 . . . . 5 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10 fssres 6585 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
118, 9, 10syl2anc 587 . . . 4 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
128freld 6551 . . . . . . 7 (𝜑 → Rel 𝐹)
13 resindm 5900 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1514eqcomd 2743 . . . . 5 (𝜑 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16 dmres 5873 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1716a1i 11 . . . . 5 (𝜑 → dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹))
1815, 17feq12d 6533 . . . 4 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
1911, 18mpbird 260 . . 3 (𝜑 → (𝐹𝐵):dom (𝐹𝐵)⟶ℝ)
2017feq2d 6531 . . 3 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
2119, 20mpbid 235 . 2 (𝜑 → (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ)
222adantr 484 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
234adantr 484 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24 simpr 488 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
2522, 23, 5, 24smfpreimalt 43939 . . . 4 ((𝜑𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹))
264dmexd 7683 . . . . . 6 (𝜑 → dom 𝐹 ∈ V)
27 elrest 16932 . . . . . 6 ((𝑆 ∈ SAlg ∧ dom 𝐹 ∈ V) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
282, 26, 27syl2anc 587 . . . . 5 (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
2928adantr 484 . . . 4 ((𝜑𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
3025, 29mpbid 235 . . 3 ((𝜑𝑎 ∈ ℝ) → ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31 elinel1 4109 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥𝐵)
3231fvresd 6737 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
3332breq1d 5063 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹𝐵)‘𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
3433rabbiia 3382 . . . . . . . . . 10 {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}
3534a1i 11 . . . . . . . . 9 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
36 rabss2 3991 . . . . . . . . . . . . 13 ((𝐵 ∩ dom 𝐹) ⊆ dom 𝐹 → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
373, 36ax-mp 5 . . . . . . . . . . . 12 {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎}
38 id 22 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
39 inss1 4143 . . . . . . . . . . . . . 14 (𝑤 ∩ dom 𝐹) ⊆ 𝑤
4039a1i 11 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) ⊆ 𝑤)
4138, 40eqsstrd 3939 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
4237, 41sstrid 3912 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
43 ssrab2 3993 . . . . . . . . . . . 12 {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝐵 ∩ dom 𝐹)
4443a1i 11 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝐵 ∩ dom 𝐹))
4542, 44ssind 4147 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46 nfrab1 3296 . . . . . . . . . . . . 13 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎}
47 nfcv 2904 . . . . . . . . . . . . 13 𝑥(𝑤 ∩ dom 𝐹)
4846, 47nfeq 2917 . . . . . . . . . . . 12 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49 elinel2 4110 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
5049adantl 485 . . . . . . . . . . . . . . . . . 18 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51 elinel1 4109 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥𝑤)
523sseli 3896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
5349, 52syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
5451, 53elind 4108 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5554adantl 485 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5638eqcomd 2743 . . . . . . . . . . . . . . . . . . . . 21 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5756adantr 484 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5855, 57eleqtrd 2840 . . . . . . . . . . . . . . . . . . 19 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
59 rabid 3290 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6059biimpi 219 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6160simprd 499 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} → (𝐹𝑥) < 𝑎)
6258, 61syl 17 . . . . . . . . . . . . . . . . . 18 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝐹𝑥) < 𝑎)
6350, 62jca 515 . . . . . . . . . . . . . . . . 17 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
64 rabid 3290 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
6563, 64sylibr 237 . . . . . . . . . . . . . . . 16 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
6665ex 416 . . . . . . . . . . . . . . 15 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
6748, 66ralrimi 3137 . . . . . . . . . . . . . 14 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
68 nfcv 2904 . . . . . . . . . . . . . . 15 𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69 nfrab1 3296 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}
7068, 69dfss3f 3891 . . . . . . . . . . . . . 14 ((𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ↔ ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7167, 70sylibr 237 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7271sseld 3900 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
7348, 72ralrimi 3137 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7473, 70sylibr 237 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7545, 74eqssd 3918 . . . . . . . . 9 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7635, 75eqtrd 2777 . . . . . . . 8 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7776adantl 485 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
78773adant2 1133 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
79223ad2ant1 1135 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80 simp1l 1199 . . . . . . . 8 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
8126, 9ssexd 5217 . . . . . . . 8 (𝜑 → (𝐵 ∩ dom 𝐹) ∈ V)
8280, 81syl 17 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝐵 ∩ dom 𝐹) ∈ V)
83 simp2 1139 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤𝑆)
84 eqid 2737 . . . . . . 7 (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
8579, 82, 83, 84elrestd 42331 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
8678, 85eqeltrd 2838 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
87863exp 1121 . . . 4 ((𝜑𝑎 ∈ ℝ) → (𝑤𝑆 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))))
8887rexlimdv 3202 . . 3 ((𝜑𝑎 ∈ ℝ) → (∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹))))
8930, 88mpd 15 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
901, 2, 7, 21, 89issmfd 43943 1 (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  {crab 3065  Vcvv 3408  cin 3865  wss 3866   cuni 4819   class class class wbr 5053  dom cdm 5551  cres 5553  Rel wrel 5556  wf 6376  cfv 6380  (class class class)co 7213  cr 10728   < clt 10867  t crest 16925  SAlgcsalg 43524  SMblFncsmblfn 43908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-pre-lttri 10803  ax-pre-lttrn 10804
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-er 8391  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-ioo 12939  df-ico 12941  df-rest 16927  df-smblfn 43909
This theorem is referenced by:  sssmfmpt  43958
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