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Theorem sssmf 41268
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 1883 . 2 𝑎𝜑
2 sssmf.s . 2 (𝜑𝑆 ∈ SAlg)
3 inss2 3867 . . 3 (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹
4 sssmf.f . . . 4 (𝜑𝐹 ∈ (SMblFn‘𝑆))
5 eqid 2651 . . . 4 dom 𝐹 = dom 𝐹
62, 4, 5smfdmss 41263 . . 3 (𝜑 → dom 𝐹 𝑆)
73, 6syl5ss 3647 . 2 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ 𝑆)
82, 4, 5smff 41262 . . . . 5 (𝜑𝐹:dom 𝐹⟶ℝ)
93a1i 11 . . . . 5 (𝜑 → (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹)
10 fssres 6108 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (𝐵 ∩ dom 𝐹) ⊆ dom 𝐹) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
118, 9, 10syl2anc 694 . . . 4 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ)
128freld 39739 . . . . . . 7 (𝜑 → Rel 𝐹)
13 resindm 5479 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
1514eqcomd 2657 . . . . 5 (𝜑 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
16 dmres 5454 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1716a1i 11 . . . . 5 (𝜑 → dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹))
1815, 17feq12d 6071 . . . 4 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹 ↾ (𝐵 ∩ dom 𝐹)):(𝐵 ∩ dom 𝐹)⟶ℝ))
1911, 18mpbird 247 . . 3 (𝜑 → (𝐹𝐵):dom (𝐹𝐵)⟶ℝ)
2017feq2d 6069 . . 3 (𝜑 → ((𝐹𝐵):dom (𝐹𝐵)⟶ℝ ↔ (𝐹𝐵):(𝐵 ∩ dom 𝐹)⟶ℝ))
2119, 20mpbid 222 . 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 41261 . . . 4 ((𝜑𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t dom 𝐹))
264dmexd 39736 . . . . . 6 (𝜑 → dom 𝐹 ∈ V)
27 elrest 16135 . . . . . 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 694 . . . . 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 222 . . 3 ((𝜑𝑎 ∈ ℝ) → ∃𝑤𝑆 {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹))
31 elinel1 3832 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥𝐵)
3231fvresd 6246 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
3332breq1d 4695 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → (((𝐹𝐵)‘𝑥) < 𝑎 ↔ (𝐹𝑥) < 𝑎))
3433rabbiia 3215 . . . . . . . . . 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 3718 . . . . . . . . . . . . 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 3866 . . . . . . . . . . . . . 14 (𝑤 ∩ dom 𝐹) ⊆ 𝑤
4039a1i 11 . . . . . . . . . . . . 13 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑤 ∩ dom 𝐹) ⊆ 𝑤)
4138, 40eqsstrd 3672 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
4237, 41syl5ss 3647 . . . . . . . . . . 11 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ 𝑤)
43 ssrab2 3720 . . . . . . . . . . . 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 3870 . . . . . . . . . 10 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} ⊆ (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
46 nfrab1 3152 . . . . . . . . . . . . 13 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎}
47 nfcv 2793 . . . . . . . . . . . . 13 𝑥(𝑤 ∩ dom 𝐹)
4846, 47nfeq 2805 . . . . . . . . . . . 12 𝑥{𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)
49 elinel2 3833 . . . . . . . . . . . . . . . . . . 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 3832 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥𝑤)
523sseli 3632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝐵 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
5349, 52syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
5451, 53elind 3831 . . . . . . . . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . 19 (({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) ∧ 𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))) → 𝑥 ∈ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎})
59 rabid 3145 . . . . . . . . . . . . . . . . . . . . 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 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 3145 . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . 15 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
6748, 66ralrimi 2986 . . . . . . . . . . . . . 14 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → ∀𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹))𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎})
68 nfcv 2793 . . . . . . . . . . . . . . 15 𝑥(𝑤 ∩ (𝐵 ∩ dom 𝐹))
69 nfrab1 3152 . . . . . . . . . . . . . . 15 𝑥{𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}
7068, 69dfss3f 3628 . . . . . . . . . . . . . 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 3635 . . . . . . . . . . . 12 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → (𝑥 ∈ (𝑤 ∩ (𝐵 ∩ dom 𝐹)) → 𝑥 ∈ {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎}))
7348, 72ralrimi 2986 . . . . . . . . . . 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 3653 . . . . . . . . 9 ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
7635, 75eqtrd 2685 . . . . . . . 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 1100 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} = (𝑤 ∩ (𝐵 ∩ dom 𝐹)))
79223ad2ant1 1102 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑆 ∈ SAlg)
80 simp1l 1105 . . . . . . . 8 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝜑)
8126, 9ssexd 4838 . . . . . . . 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 1082 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → 𝑤𝑆)
84 eqid 2651 . . . . . . 7 (𝑤 ∩ (𝐵 ∩ dom 𝐹)) = (𝑤 ∩ (𝐵 ∩ dom 𝐹))
8579, 82, 83, 84elrestd 39605 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → (𝑤 ∩ (𝐵 ∩ dom 𝐹)) ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
8678, 85eqeltrd 2730 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ 𝑤𝑆 ∧ {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹)) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))
87863exp 1283 . . . 4 ((𝜑𝑎 ∈ ℝ) → (𝑤𝑆 → ({𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) < 𝑎} = (𝑤 ∩ dom 𝐹) → {𝑥 ∈ (𝐵 ∩ dom 𝐹) ∣ ((𝐹𝐵)‘𝑥) < 𝑎} ∈ (𝑆t (𝐵 ∩ dom 𝐹)))))
8887rexlimdv 3059 . . 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 41265 1 (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607   cuni 4468   class class class wbr 4685  dom cdm 5143  cres 5145  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  cr 9973   < clt 10112  t crest 16128  SAlgcsalg 40846  SMblFncsmblfn 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-ioo 12217  df-ico 12219  df-rest 16130  df-smblfn 41231
This theorem is referenced by:  sssmfmpt  41280
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