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Theorem issmflem 42881
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmflem.s (𝜑𝑆 ∈ SAlg)
issmflem.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmflem (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎,𝑥   𝜑,𝑎,𝑥

Proof of Theorem issmflem
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2 df-smblfn 42855 . . . . . . . . 9 SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
3 unieq 4838 . . . . . . . . . . . 12 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 7161 . . . . . . . . . . 11 (𝑠 = 𝑆 → (ℝ ↑pm 𝑠) = (ℝ ↑pm 𝑆))
54rabeqdv 3482 . . . . . . . . . 10 (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
6 oveq1 7152 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (𝑠t dom 𝑓) = (𝑆t dom 𝑓))
76eleq2d 2895 . . . . . . . . . . . 12 (𝑠 = 𝑆 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
87ralbidv 3194 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
98rabbidv 3478 . . . . . . . . . 10 (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
105, 9eqtrd 2853 . . . . . . . . 9 (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
11 issmflem.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
12 ovex 7178 . . . . . . . . . . 11 (ℝ ↑pm 𝑆) ∈ V
1312rabex 5226 . . . . . . . . . 10 {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ∈ V
1413a1i 11 . . . . . . . . 9 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ∈ V)
152, 10, 11, 14fvmptd3 6783 . . . . . . . 8 (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
1615adantr 481 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
171, 16eleqtrd 2912 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
18 elrabi 3672 . . . . . 6 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} → 𝐹 ∈ (ℝ ↑pm 𝑆))
1917, 18syl 17 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
20 issmflem.d . . . . . . 7 𝐷 = dom 𝐹
21 elpmi2 41365 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → dom 𝐹 𝑆)
2220, 21eqsstrid 4012 . . . . . 6 (𝐹 ∈ (ℝ ↑pm 𝑆) → 𝐷 𝑆)
2322adantl 482 . . . . 5 ((𝜑𝐹 ∈ (ℝ ↑pm 𝑆)) → 𝐷 𝑆)
2419, 23syldan 591 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
25 elpmi 8414 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 𝑆))
2619, 25syl 17 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 𝑆))
2726simpld 495 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
2820feq2i 6499 . . . . . 6 (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ)
2928a1i 11 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ))
3027, 29mpbird 258 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
31 cnveq 5737 . . . . . . . . . . . . . 14 (𝑓 = 𝐹𝑓 = 𝐹)
3231imaeq1d 5921 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓 “ (-∞(,)𝑎)) = (𝐹 “ (-∞(,)𝑎)))
33 dmeq 5765 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3433oveq2d 7161 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑆t dom 𝑓) = (𝑆t dom 𝐹))
3532, 34eleq12d 2904 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3635ralbidv 3194 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3736elrab 3677 . . . . . . . . . 10 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ↔ (𝐹 ∈ (ℝ ↑pm 𝑆) ∧ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3837simprbi 497 . . . . . . . . 9 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
3917, 38syl 17 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4039adantr 481 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
41 simpr 485 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
42 rspa 3203 . . . . . . 7 ((∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4340, 41, 42syl2anc 584 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4430adantr 481 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
45 simpl 483 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
46 simpr 485 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
4746rexrd 10679 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
4845, 47preimaioomnf 42874 . . . . . . . . 9 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎})
4948eqcomd 2824 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5044, 41, 49syl2anc 584 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5120oveq2i 7156 . . . . . . . 8 (𝑆t 𝐷) = (𝑆t dom 𝐹)
5251a1i 11 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
5350, 52eleq12d 2904 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
5443, 53mpbird 258 . . . . 5 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5554ralrimiva 3179 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5624, 30, 553jca 1120 . . 3 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
5756ex 413 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
58 reex 10616 . . . . . . . . 9 ℝ ∈ V
5958a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
6011uniexd 7457 . . . . . . . . 9 (𝜑 𝑆 ∈ V)
6160adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝑆 ∈ V)
62 simprr 769 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
63 fssxp 6527 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
6463adantl 482 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
65 xpss1 5567 . . . . . . . . . . . 12 (𝐷 𝑆 → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6665adantr 481 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6764, 66sstrd 3974 . . . . . . . . . 10 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ ( 𝑆 × ℝ))
6867adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ ( 𝑆 × ℝ))
69 dmss 5764 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 ⊆ dom ( 𝑆 × ℝ))
70 dmxpss 6021 . . . . . . . . . . . . 13 dom ( 𝑆 × ℝ) ⊆ 𝑆
7170a1i 11 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom ( 𝑆 × ℝ) ⊆ 𝑆)
7269, 71sstrd 3974 . . . . . . . . . . 11 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 𝑆)
7372adantl 482 . . . . . . . . . 10 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → dom 𝐹 𝑆)
7420, 73eqsstrid 4012 . . . . . . . . 9 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → 𝐷 𝑆)
7568, 74syldan 591 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐷 𝑆)
76 elpm2r 8413 . . . . . . . 8 (((ℝ ∈ V ∧ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7759, 61, 62, 75, 76syl22anc 834 . . . . . . 7 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
78773adantr3 1163 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7920a1i 11 . . . . . . . . . . . . 13 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐷 = dom 𝐹)
8079oveq2d 7161 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
8149, 80eleq12d 2904 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8281ralbidva 3193 . . . . . . . . . 10 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8382biimpd 230 . . . . . . . . 9 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8483imp 407 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8584adantl 482 . . . . . . 7 ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
86853adantr1 1161 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8778, 86jca 512 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → (𝐹 ∈ (ℝ ↑pm 𝑆) ∧ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8887, 37sylibr 235 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
8915eqcomd 2824 . . . . 5 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9089adantr 481 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9188, 90eleqtrd 2912 . . 3 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
9291ex 413 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
9357, 92impbid 213 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  wss 3933   cuni 4830   class class class wbr 5057   × cxp 5546  ccnv 5547  dom cdm 5548  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cr 10524  -∞cmnf 10661   < clt 10663  (,)cioo 12726  t crest 16682  SAlgcsalg 42470  SMblFncsmblfn 42854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-ioo 12730  df-ico 12732  df-smblfn 42855
This theorem is referenced by:  issmf  42882
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