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Theorem issmflem 41460
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 479 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2 df-smblfn 41434 . . . . . . . . . 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 4596 . . . . . . . . . . . . 13 (𝑠 = 𝑆 𝑠 = 𝑆)
54oveq2d 6830 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (ℝ ↑pm 𝑠) = (ℝ ↑pm 𝑆))
65rabeqd 39793 . . . . . . . . . . 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 6821 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠t dom 𝑓) = (𝑆t dom 𝑓))
87eleq2d 2825 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
98ralbidv 3124 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
109rabbidv 3329 . . . . . . . . . . 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 2794 . . . . . . . . . 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 473 . . . . . . . . 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 6842 . . . . . . . . . . 11 (ℝ ↑pm 𝑆) ∈ V
1514rabex 4964 . . . . . . . . . 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 6451 . . . . . . . 8 (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
1817adantr 472 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
191, 18eleqtrd 2841 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
20 elrabi 3499 . . . . . 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 39935 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → dom 𝐹 𝑆)
2422, 23syl5eqss 3790 . . . . . 6 (𝐹 ∈ (ℝ ↑pm 𝑆) → 𝐷 𝑆)
2524adantl 473 . . . . 5 ((𝜑𝐹 ∈ (ℝ ↑pm 𝑆)) → 𝐷 𝑆)
2621, 25syldan 488 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
27 elpmi 8044 . . . . . . 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 477 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
3022feq2i 6198 . . . . . 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 5451 . . . . . . . . . . . . . 14 (𝑓 = 𝐹𝑓 = 𝐹)
3433imaeq1d 5623 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓 “ (-∞(,)𝑎)) = (𝐹 “ (-∞(,)𝑎)))
35 dmeq 5479 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3635oveq2d 6830 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑆t dom 𝑓) = (𝑆t dom 𝐹))
3734, 36eleq12d 2833 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3837ralbidv 3124 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3938elrab 3504 . . . . . . . . . 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 483 . . . . . . . . 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 472 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
43 simpr 479 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
44 rspa 3068 . . . . . . 7 ((∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4542, 43, 44syl2anc 696 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4632adantr 472 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
47 simpl 474 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
48 simpr 479 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
4948rexrd 10301 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
5047, 49preimaioomnf 41453 . . . . . . . . 9 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎})
5150eqcomd 2766 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5246, 43, 51syl2anc 696 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5322oveq2i 6825 . . . . . . . 8 (𝑆t 𝐷) = (𝑆t dom 𝐹)
5453a1i 11 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
5552, 54eleq12d 2833 . . . . . 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 3104 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5826, 32, 573jca 1123 . . 3 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)))
5958ex 449 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))
60 reex 10239 . . . . . . . . 9 ℝ ∈ V
6160a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
6213uniexd 39798 . . . . . . . . 9 (𝜑 𝑆 ∈ V)
6362adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝑆 ∈ V)
64 simprr 813 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
65 fssxp 6221 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
6665adantl 473 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
67 xpss1 5284 . . . . . . . . . . . 12 (𝐷 𝑆 → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6867adantr 472 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6966, 68sstrd 3754 . . . . . . . . . 10 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ ( 𝑆 × ℝ))
7069adantl 473 . . . . . . . . 9 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ ( 𝑆 × ℝ))
71 dmss 5478 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 ⊆ dom ( 𝑆 × ℝ))
72 dmxpss 5723 . . . . . . . . . . . . 13 dom ( 𝑆 × ℝ) ⊆ 𝑆
7372a1i 11 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom ( 𝑆 × ℝ) ⊆ 𝑆)
7471, 73sstrd 3754 . . . . . . . . . . 11 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 𝑆)
7574adantl 473 . . . . . . . . . 10 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → dom 𝐹 𝑆)
7622, 75syl5eqss 3790 . . . . . . . . 9 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → 𝐷 𝑆)
7770, 76syldan 488 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐷 𝑆)
78 elpm2r 8043 . . . . . . . 8 (((ℝ ∈ V ∧ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7961, 63, 64, 77, 78syl22anc 1478 . . . . . . 7 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
80793adantr3 1177 . . . . . 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 6830 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
8351, 82eleq12d 2833 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8483ralbidva 3123 . . . . . . . . . 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 444 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷)) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8786adantl 473 . . . . . . 7 ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
88873adantr1 1175 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8980, 88jca 555 . . . . 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 2766 . . . . 5 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9291adantr 472 . . . 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 2841 . . 3 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
9493ex 449 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wss 3715   cuni 4588   class class class wbr 4804  cmpt 4881   × cxp 5264  ccnv 5265  dom cdm 5266  cima 5269  wf 6045  cfv 6049  (class class class)co 6814  pm cpm 8026  cr 10147  -∞cmnf 10284   < clt 10286  (,)cioo 12388  t crest 16303  SAlgcsalg 41049  SMblFncsmblfn 41433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-pre-lttri 10222  ax-pre-lttrn 10223
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-er 7913  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-ioo 12392  df-ico 12394  df-smblfn 41434
This theorem is referenced by:  issmf  41461
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