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Theorem issmflem 46747
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 484 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2 df-smblfn 46716 . . . . . . . . 9 SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
3 unieq 4917 . . . . . . . . . . . 12 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 7448 . . . . . . . . . . 11 (𝑠 = 𝑆 → (ℝ ↑pm 𝑠) = (ℝ ↑pm 𝑆))
54rabeqdv 3451 . . . . . . . . . 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 7439 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (𝑠t dom 𝑓) = (𝑆t dom 𝑓))
76eleq2d 2826 . . . . . . . . . . . 12 (𝑠 = 𝑆 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
87ralbidv 3177 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
98rabbidv 3443 . . . . . . . . . 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 2776 . . . . . . . . 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 7465 . . . . . . . . . . 11 (ℝ ↑pm 𝑆) ∈ V
1312rabex 5338 . . . . . . . . . 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 7038 . . . . . . . 8 (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
1615adantr 480 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
171, 16eleqtrd 2842 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
18 elrabi 3686 . . . . . 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 45235 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → dom 𝐹 𝑆)
2220, 21eqsstrid 4021 . . . . . 6 (𝐹 ∈ (ℝ ↑pm 𝑆) → 𝐷 𝑆)
2322adantl 481 . . . . 5 ((𝜑𝐹 ∈ (ℝ ↑pm 𝑆)) → 𝐷 𝑆)
2419, 23syldan 591 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
25 elpmi 8887 . . . . . . 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 494 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
2820feq2i 6727 . . . . . 6 (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ)
2928a1i 11 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ))
3027, 29mpbird 257 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
31 cnveq 5883 . . . . . . . . . . . . . 14 (𝑓 = 𝐹𝑓 = 𝐹)
3231imaeq1d 6076 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓 “ (-∞(,)𝑎)) = (𝐹 “ (-∞(,)𝑎)))
33 dmeq 5913 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3433oveq2d 7448 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑆t dom 𝑓) = (𝑆t dom 𝐹))
3532, 34eleq12d 2834 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3635ralbidv 3177 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3736elrab 3691 . . . . . . . . . 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 496 . . . . . . . . 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 480 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
41 simpr 484 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
42 rspa 3247 . . . . . . 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 480 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
45 simpl 482 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
46 simpr 484 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
4746rexrd 11312 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
4845, 47preimaioomnf 46739 . . . . . . . . 9 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎})
4948eqcomd 2742 . . . . . . . 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 7443 . . . . . . . 8 (𝑆t 𝐷) = (𝑆t dom 𝐹)
5251a1i 11 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
5350, 52eleq12d 2834 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
5443, 53mpbird 257 . . . . 5 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5554ralrimiva 3145 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5624, 30, 553jca 1128 . . 3 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
5756ex 412 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
58 reex 11247 . . . . . . . . 9 ℝ ∈ V
5958a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
6011uniexd 7763 . . . . . . . . 9 (𝜑 𝑆 ∈ V)
6160adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝑆 ∈ V)
62 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
63 fssxp 6762 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
6463adantl 481 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
65 xpss1 5703 . . . . . . . . . . . 12 (𝐷 𝑆 → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6665adantr 480 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6764, 66sstrd 3993 . . . . . . . . . 10 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ ( 𝑆 × ℝ))
6867adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ ( 𝑆 × ℝ))
69 dmss 5912 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 ⊆ dom ( 𝑆 × ℝ))
70 dmxpss 6190 . . . . . . . . . . . . 13 dom ( 𝑆 × ℝ) ⊆ 𝑆
7170a1i 11 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom ( 𝑆 × ℝ) ⊆ 𝑆)
7269, 71sstrd 3993 . . . . . . . . . . 11 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 𝑆)
7372adantl 481 . . . . . . . . . 10 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → dom 𝐹 𝑆)
7420, 73eqsstrid 4021 . . . . . . . . 9 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → 𝐷 𝑆)
7568, 74syldan 591 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐷 𝑆)
76 elpm2r 8886 . . . . . . . 8 (((ℝ ∈ V ∧ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7759, 61, 62, 75, 76syl22anc 838 . . . . . . 7 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
78773adantr3 1171 . . . . . 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 7448 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
8149, 80eleq12d 2834 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8281ralbidva 3175 . . . . . . . . . 10 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8382biimpd 229 . . . . . . . . 9 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8483imp 406 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8584adantl 481 . . . . . . 7 ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
86853adantr1 1169 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8778, 86jca 511 . . . . 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 234 . . . 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 2742 . . . . 5 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9089adantr 480 . . . 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 2842 . . 3 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
9291ex 412 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
9357, 92impbid 212 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  wss 3950   cuni 4906   class class class wbr 5142   × cxp 5682  ccnv 5683  dom cdm 5684  cima 5687  wf 6556  cfv 6560  (class class class)co 7432  pm cpm 8868  cr 11155  -∞cmnf 11294   < clt 11296  (,)cioo 13388  t crest 17466  SAlgcsalg 46328  SMblFncsmblfn 46715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-pre-lttri 11230  ax-pre-lttrn 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-er 8746  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-ioo 13392  df-ico 13394  df-smblfn 46716
This theorem is referenced by:  issmf  46748
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