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Theorem issmflem 40243
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 477 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2 df-smblfn 40217 . . . . . . . . . 10 SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
32a1i 11 . . . . . . . . 9 (𝜑 → SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)}))
4 unieq 4410 . . . . . . . . . . . . 13 (𝑠 = 𝑆 𝑠 = 𝑆)
54oveq2d 6620 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (ℝ ↑pm 𝑠) = (ℝ ↑pm 𝑆))
65rabeqd 38761 . . . . . . . . . . 11 (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
7 oveq1 6611 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠t dom 𝑓) = (𝑆t dom 𝑓))
87eleq2d 2684 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
98ralbidv 2980 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
109rabbidv 3177 . . . . . . . . . . 11 (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
116, 10eqtrd 2655 . . . . . . . . . 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)})
1211adantl 482 . . . . . . . . 9 ((𝜑𝑠 = 𝑆) → {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
13 issmflem.s . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
14 ovex 6632 . . . . . . . . . . 11 (ℝ ↑pm 𝑆) ∈ V
1514rabex 4773 . . . . . . . . . 10 {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ∈ V
1615a1i 11 . . . . . . . . 9 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ∈ V)
173, 12, 13, 16fvmptd 6245 . . . . . . . 8 (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
1817adantr 481 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
191, 18eleqtrd 2700 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
20 elrabi 3342 . . . . . 6 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} → 𝐹 ∈ (ℝ ↑pm 𝑆))
2119, 20syl 17 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
22 issmflem.d . . . . . . 7 𝐷 = dom 𝐹
23 elpmi2 38892 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → dom 𝐹 𝑆)
2422, 23syl5eqss 3628 . . . . . 6 (𝐹 ∈ (ℝ ↑pm 𝑆) → 𝐷 𝑆)
2524adantl 482 . . . . 5 ((𝜑𝐹 ∈ (ℝ ↑pm 𝑆)) → 𝐷 𝑆)
2621, 25syldan 487 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
27 elpmi 7820 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 𝑆))
2821, 27syl 17 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 𝑆))
2928simpld 475 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
3022feq2i 5994 . . . . . 6 (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ)
3130a1i 11 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ))
3229, 31mpbird 247 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
33 cnveq 5256 . . . . . . . . . . . . . 14 (𝑓 = 𝐹𝑓 = 𝐹)
3433imaeq1d 5424 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓 “ (-∞(,)𝑎)) = (𝐹 “ (-∞(,)𝑎)))
35 dmeq 5284 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3635oveq2d 6620 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑆t dom 𝑓) = (𝑆t dom 𝐹))
3734, 36eleq12d 2692 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3837ralbidv 2980 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3938elrab 3346 . . . . . . . . . 10 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} ↔ (𝐹 ∈ (ℝ ↑pm 𝑆) ∧ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
4039simprbi 480 . . . . . . . . 9 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4119, 40syl 17 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4241adantr 481 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
43 simpr 477 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
44 rspa 2925 . . . . . . 7 ((∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4542, 43, 44syl2anc 692 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4632adantr 481 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
47 simpl 473 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
48 simpr 477 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
4948rexrd 10033 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
5047, 49preimaioomnf 40236 . . . . . . . . 9 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎})
5150eqcomd 2627 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5246, 43, 51syl2anc 692 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5322oveq2i 6615 . . . . . . . 8 (𝑆t 𝐷) = (𝑆t dom 𝐹)
5453a1i 11 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
5552, 54eleq12d 2692 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
5645, 55mpbird 247 . . . . 5 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5756ralrimiva 2960 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5826, 32, 573jca 1240 . . 3 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
5958ex 450 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
60 reex 9971 . . . . . . . . 9 ℝ ∈ V
6160a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
6213uniexd 38766 . . . . . . . . 9 (𝜑 𝑆 ∈ V)
6362adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝑆 ∈ V)
64 simprr 795 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
65 fssxp 6017 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
6665adantl 482 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
67 xpss1 5189 . . . . . . . . . . . 12 (𝐷 𝑆 → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6867adantr 481 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6966, 68sstrd 3593 . . . . . . . . . 10 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ ( 𝑆 × ℝ))
7069adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ ( 𝑆 × ℝ))
71 dmss 5283 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 ⊆ dom ( 𝑆 × ℝ))
72 dmxpss 5524 . . . . . . . . . . . . 13 dom ( 𝑆 × ℝ) ⊆ 𝑆
7372a1i 11 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom ( 𝑆 × ℝ) ⊆ 𝑆)
7471, 73sstrd 3593 . . . . . . . . . . 11 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 𝑆)
7574adantl 482 . . . . . . . . . 10 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → dom 𝐹 𝑆)
7622, 75syl5eqss 3628 . . . . . . . . 9 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → 𝐷 𝑆)
7770, 76syldan 487 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐷 𝑆)
78 elpm2r 7819 . . . . . . . 8 (((ℝ ∈ V ∧ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7961, 63, 64, 77, 78syl22anc 1324 . . . . . . 7 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
80793adantr3 1220 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (ℝ ↑pm 𝑆))
8122a1i 11 . . . . . . . . . . . . 13 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐷 = dom 𝐹)
8281oveq2d 6620 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
8351, 82eleq12d 2692 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8483ralbidva 2979 . . . . . . . . . 10 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8584biimpd 219 . . . . . . . . 9 (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8685imp 445 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8786adantl 482 . . . . . . 7 ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
88873adantr1 1218 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8980, 88jca 554 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → (𝐹 ∈ (ℝ ↑pm 𝑆) ∧ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
9089, 39sylibr 224 . . . 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)})
9117eqcomd 2627 . . . . 5 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9291adantr 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))
9390, 92eleqtrd 2700 . . 3 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
9493ex 450 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
9559, 94impbid 202 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555   cuni 4402   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  cr 9879  -∞cmnf 10016   < clt 10018  (,)cioo 12117  t crest 16002  SAlgcsalg 39835  SMblFncsmblfn 40216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-ioo 12121  df-ico 12123  df-smblfn 40217
This theorem is referenced by:  issmf  40244
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