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Theorem sssmf 41467
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 1995 . 2 𝑎𝜑
2 sssmf.s . 2 (𝜑𝑆 ∈ SAlg)
3 inss2 3982 . . 3 (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4 sssmf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
5 eqid 2771 . . . 4 dom 𝐹 = dom 𝐹
62, 4, 5smfdmss 41462 . . 3 (𝜑 → dom 𝐹 𝑆)
73, 6syl5ss 3763 . 2 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ 𝑆)
82, 4, 5smff 41461 . . . . 5 (𝜑𝐹:dom 𝐹⟶ℝ)
93a1i 11 . . . . 5 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10 fssres 6210 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
118, 9, 10syl2anc 565 . . . 4 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
128freld 39943 . . . . . . 7 (𝜑 → Rel 𝐹)
13 resindm 5585 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1514eqcomd 2777 . . . . 5 (𝜑 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16 dmres 5560 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1716a1i 11 . . . . 5 (𝜑 → dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹))
1815, 17feq12d 6173 . . . 4 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
1911, 18mpbird 247 . . 3 (𝜑 → (𝐹𝐵):dom (𝐹𝐵)⟶ℝ)
2017feq2d 6171 . . 3 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
2119, 20mpbid 222 . 2 (𝜑 → (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ)
222adantr 466 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
234adantr 466 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
24 simpr 471 . . . . 5 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
2522, 23, 5, 24smfpreimalt 41460 . . . 4 ((𝜑𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹))
264dmexd 39940 . . . . . 6 (𝜑 → dom 𝐹 ∈ V)
27 elrest 16296 . . . . . 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 565 . . . . 5 (𝜑 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
2928adantr 466 . . . 4 ((𝜑𝑎 ∈ ℝ) → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹) ↔ ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)))
3025, 29mpbid 222 . . 3 ((𝜑𝑎 ∈ ℝ) → ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31 elinel1 3950 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥𝐵)
3231fvresd 6349 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
3332breq1d 4796 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹𝐵)‘𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
3433rabbiia 3334 . . . . . . . . . 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 3834 . . . . . . . . . . . . 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 3981 . . . . . . . . . . . . . 14 (𝑤 ∩ dom 𝐹) ⊆ 𝑤
4039a1i 11 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) ⊆ 𝑤)
4138, 40eqsstrd 3788 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
4237, 41syl5ss 3763 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
43 ssrab2 3836 . . . . . . . . . . . 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 3985 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46 nfrab1 3271 . . . . . . . . . . . . 13 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎}
47 nfcv 2913 . . . . . . . . . . . . 13 𝑥(𝑤 ∩ dom 𝐹)
4846, 47nfeq 2925 . . . . . . . . . . . 12 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49 elinel2 3951 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
5049adantl 467 . . . . . . . . . . . . . . . . . 18 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝐵 ∩ dom 𝐹))
51 elinel1 3950 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥𝑤)
523sseli 3748 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
5349, 52syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
5451, 53elind 3949 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5554adantl 467 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ (𝑤 ∩ dom 𝐹))
5638eqcomd 2777 . . . . . . . . . . . . . . . . . . . . 21 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5756adantr 466 . . . . . . . . . . . . . . . . . . . 20 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑤 ∩ dom 𝐹) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
5855, 57eleqtrd 2852 . . . . . . . . . . . . . . . . . . 19 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
59 rabid 3264 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6059biimpi 206 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) < 𝑎))
6160simprd 477 . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . 17 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
64 rabid 3264 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ (𝐹𝑥) < 𝑎))
6563, 64sylibr 224 . . . . . . . . . . . . . . . 16 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
6665ex 397 . . . . . . . . . . . . . . 15 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
6748, 66ralrimi 3106 . . . . . . . . . . . . . 14 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
68 nfcv 2913 . . . . . . . . . . . . . . 15 𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69 nfrab1 3271 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}
7068, 69dfss3f 3744 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7271sseld 3751 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
7348, 72ralrimi 3106 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7473, 70sylibr 224 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ⊆ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
7545, 74eqssd 3769 . . . . . . . . 9 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7635, 75eqtrd 2805 . . . . . . . 8 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7776adantl 467 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
78773adant2 1125 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
79223ad2ant1 1127 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80 simp1l 1239 . . . . . . . 8 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
8126, 9ssexd 4939 . . . . . . . 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 1131 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤𝑆)
84 eqid 2771 . . . . . . 7 (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
8579, 82, 83, 84elrestd 39812 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
8678, 85eqeltrd 2850 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
87863exp 1112 . . . 4 ((𝜑𝑎 ∈ ℝ) → (𝑤𝑆 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))))
8887rexlimdv 3178 . . 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 41464 1 (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cin 3722  wss 3723   cuni 4574   class class class wbr 4786  dom cdm 5249  cres 5251  Rel wrel 5254  wf 6027  cfv 6031  (class class class)co 6793  cr 10137   < clt 10276  t crest 16289  SAlgcsalg 41045  SMblFncsmblfn 41429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-ioo 12384  df-ico 12386  df-rest 16291  df-smblfn 41430
This theorem is referenced by:  sssmfmpt  41479
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