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Theorem sssmf 39625
 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 1830 . 2 𝑎𝜑
2 sssmf.s . 2 (𝜑𝑆 ∈ SAlg)
3 inss2 3796 . . 3 (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4 sssmf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
5 eqid 2610 . . . 4 dom 𝐹 = dom 𝐹
62, 4, 5smfdmss 39619 . . 3 (𝜑 → dom 𝐹 𝑆)
73, 6syl5ss 3579 . 2 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ 𝑆)
82, 4, 5smff 39618 . . . . 5 (𝜑𝐹:dom 𝐹⟶ℝ)
93a1i 11 . . . . 5 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10 fssres 5983 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
118, 9, 10syl2anc 691 . . . 4 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
128freld 38420 . . . . . . 7 (𝜑 → Rel 𝐹)
13 resindm 5364 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1514eqcomd 2616 . . . . 5 (𝜑 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16 dmres 5339 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1716a1i 11 . . . . 5 (𝜑 → dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹))
1815, 17feq12d 5946 . . . 4 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
1911, 18mpbird 246 . . 3 (𝜑 → (𝐹𝐵):dom (𝐹𝐵)⟶ℝ)
2017feq2d 5944 . . 3 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
2119, 20mpbid 221 . 2 (𝜑 → (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ)
222adantr 480 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
234adantr 480 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24 simpr 476 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
2522, 23, 5, 24smfpreimalt 39617 . . . 4 ((𝜑𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹))
264dmexd 38417 . . . . . 6 (𝜑 → dom 𝐹 ∈ V)
27 elrest 15911 . . . . . 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 691 . . . . 5 (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
2928adantr 480 . . . 4 ((𝜑𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
3025, 29mpbid 221 . . 3 ((𝜑𝑎 ∈ ℝ) → ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31 elinel1 3761 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥𝐵)
3231fvresd 6118 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
3332breq1d 4593 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹𝐵)‘𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
3433rabbiia 3161 . . . . . . . . . 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 3648 . . . . . . . . . . . . 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 3795 . . . . . . . . . . . . . 14 (𝑤 ∩ dom 𝐹) ⊆ 𝑤
4039a1i 11 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) ⊆ 𝑤)
4138, 40eqsstrd 3602 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
4237, 41syl5ss 3579 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
43 ssrab2 3650 . . . . . . . . . . . 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 3799 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46 nfrab1 3099 . . . . . . . . . . . . 13 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎}
47 nfcv 2751 . . . . . . . . . . . . 13 𝑥(𝑤 ∩ dom 𝐹)
4846, 47nfeq 2762 . . . . . . . . . . . 12 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49 elinel2 3762 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
5049adantl 481 . . . . . . . . . . . . . . . . . 18 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51 elinel1 3761 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥𝑤)
523sseli 3564 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
5349, 52syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
5451, 53elind 3760 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5554adantl 481 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5638eqcomd 2616 . . . . . . . . . . . . . . . . . . . . 21 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5756adantr 480 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5855, 57eleqtrd 2690 . . . . . . . . . . . . . . . . . . 19 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
59 rabid 3095 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6059biimpi 205 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6160simprd 478 . . . . . . . . . . . . . . . . . . 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 553 . . . . . . . . . . . . . . . . 17 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
64 rabid 3095 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
6563, 64sylibr 223 . . . . . . . . . . . . . . . 16 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
6665ex 449 . . . . . . . . . . . . . . 15 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
6748, 66ralrimi 2940 . . . . . . . . . . . . . 14 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
68 nfcv 2751 . . . . . . . . . . . . . . 15 𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69 nfrab1 3099 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}
7068, 69dfss3f 3560 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7271sseld 3567 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
7348, 72ralrimi 2940 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7473, 70sylibr 223 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7545, 74eqssd 3585 . . . . . . . . 9 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7635, 75eqtrd 2644 . . . . . . . 8 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7776adantl 481 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
78773adant2 1073 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
79223ad2ant1 1075 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80 simp1l 1078 . . . . . . . 8 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
8126, 9ssexd 4733 . . . . . . . 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 1055 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤𝑆)
84 eqid 2610 . . . . . . 7 (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
8579, 82, 83, 84elrestd 38322 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
8678, 85eqeltrd 2688 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
87863exp 1256 . . . 4 ((𝜑𝑎 ∈ ℝ) → (𝑤𝑆 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))))
8887rexlimdv 3012 . . 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 39621 1 (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   < clt 9953   ↾t crest 15904  SAlgcsalg 39204  SMblFncsmblfn 39586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ioo 12050  df-ico 12052  df-rest 15906  df-smblfn 39587 This theorem is referenced by:  sssmfmpt  39637
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