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Theorem issmflem 41257
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 476 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2 df-smblfn 41231 . . . . . . . . . 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 4476 . . . . . . . . . . . . 13 (𝑠 = 𝑆 𝑠 = 𝑆)
54oveq2d 6706 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (ℝ ↑pm 𝑠) = (ℝ ↑pm 𝑆))
65rabeqd 39590 . . . . . . . . . . 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 6697 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠t dom 𝑓) = (𝑆t dom 𝑓))
87eleq2d 2716 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
98ralbidv 3015 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)))
109rabbidv 3220 . . . . . . . . . . 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 2685 . . . . . . . . . 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 481 . . . . . . . . 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 6718 . . . . . . . . . . 11 (ℝ ↑pm 𝑆) ∈ V
1514rabex 4845 . . . . . . . . . 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 6327 . . . . . . . 8 (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
1817adantr 480 . . . . . . 7 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
191, 18eleqtrd 2732 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)})
20 elrabi 3391 . . . . . 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 39732 . . . . . . 7 (𝐹 ∈ (ℝ ↑pm 𝑆) → dom 𝐹 𝑆)
2422, 23syl5eqss 3682 . . . . . 6 (𝐹 ∈ (ℝ ↑pm 𝑆) → 𝐷 𝑆)
2524adantl 481 . . . . 5 ((𝜑𝐹 ∈ (ℝ ↑pm 𝑆)) → 𝐷 𝑆)
2621, 25syldan 486 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
27 elpmi 7918 . . . . . . 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 474 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
3022feq2i 6075 . . . . . 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 5328 . . . . . . . . . . . . . 14 (𝑓 = 𝐹𝑓 = 𝐹)
3433imaeq1d 5500 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓 “ (-∞(,)𝑎)) = (𝐹 “ (-∞(,)𝑎)))
35 dmeq 5356 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3635oveq2d 6706 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑆t dom 𝑓) = (𝑆t dom 𝐹))
3734, 36eleq12d 2724 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3837ralbidv 3015 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
3938elrab 3396 . . . . . . . . . 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 479 . . . . . . . . 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 480 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
43 simpr 476 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
44 rspa 2959 . . . . . . 7 ((∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4542, 43, 44syl2anc 694 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
4632adantr 480 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
47 simpl 472 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
48 simpr 476 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
4948rexrd 10127 . . . . . . . . . 10 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
5047, 49preimaioomnf 41250 . . . . . . . . 9 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝐹 “ (-∞(,)𝑎)) = {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎})
5150eqcomd 2657 . . . . . . . 8 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5246, 43, 51syl2anc 694 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} = (𝐹 “ (-∞(,)𝑎)))
5322oveq2i 6701 . . . . . . . 8 (𝑆t 𝐷) = (𝑆t dom 𝐹)
5453a1i 11 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
5552, 54eleq12d 2724 . . . . . 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 2995 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))
5826, 32, 573jca 1261 . . 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 10065 . . . . . . . . 9 ℝ ∈ V
6160a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
6213uniexd 39595 . . . . . . . . 9 (𝜑 𝑆 ∈ V)
6362adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝑆 ∈ V)
64 simprr 811 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
65 fssxp 6098 . . . . . . . . . . . 12 (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
6665adantl 481 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
67 xpss1 5161 . . . . . . . . . . . 12 (𝐷 𝑆 → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6867adantr 480 . . . . . . . . . . 11 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ ( 𝑆 × ℝ))
6966, 68sstrd 3646 . . . . . . . . . 10 ((𝐷 𝑆𝐹:𝐷⟶ℝ) → 𝐹 ⊆ ( 𝑆 × ℝ))
7069adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ ( 𝑆 × ℝ))
71 dmss 5355 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 ⊆ dom ( 𝑆 × ℝ))
72 dmxpss 5600 . . . . . . . . . . . . 13 dom ( 𝑆 × ℝ) ⊆ 𝑆
7372a1i 11 . . . . . . . . . . . 12 (𝐹 ⊆ ( 𝑆 × ℝ) → dom ( 𝑆 × ℝ) ⊆ 𝑆)
7471, 73sstrd 3646 . . . . . . . . . . 11 (𝐹 ⊆ ( 𝑆 × ℝ) → dom 𝐹 𝑆)
7574adantl 481 . . . . . . . . . 10 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → dom 𝐹 𝑆)
7622, 75syl5eqss 3682 . . . . . . . . 9 ((𝜑𝐹 ⊆ ( 𝑆 × ℝ)) → 𝐷 𝑆)
7770, 76syldan 486 . . . . . . . 8 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐷 𝑆)
78 elpm2r 7917 . . . . . . . 8 (((ℝ ∈ V ∧ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 𝑆)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
7961, 63, 64, 77, 78syl22anc 1367 . . . . . . 7 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm 𝑆))
80793adantr3 1242 . . . . . 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 6706 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆t 𝐷) = (𝑆t dom 𝐹))
8351, 82eleq12d 2724 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷) ↔ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹)))
8483ralbidva 3014 . . . . . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
88873adantr1 1240 . . . . . 6 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))) → ∀𝑎 ∈ ℝ (𝐹 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝐹))
8980, 88jca 553 . . . . 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 2657 . . . . 5 (𝜑 → {𝑓 ∈ (ℝ ↑pm 𝑆) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑆t dom 𝑓)} = (SMblFn‘𝑆))
9291adantr 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))
9390, 92eleqtrd 2732 . . 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 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wss 3607   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  pm cpm 7900  cr 9973  -∞cmnf 10110   < clt 10112  (,)cioo 12213  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-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-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-smblfn 41231
This theorem is referenced by:  issmf  41258
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