Theorem smfpreimalt 39617

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfpreimalt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpreimalt.f	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
smfpreimalt.d	⊢ 𝐷 = dom 𝐹
smfpreimalt.a	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
smfpreimalt	⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝑥,𝐹

Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)

Proof of Theorem smfpreimalt

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpreimalt.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		smfpreimalt.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
3		smfpreimalt.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
4		smfpreimalt.d	. . . . 5 ⊢ 𝐷 = dom 𝐹
5	3, 4	issmf 39614	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
6	2, 5	mpbid 221	. . 3 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
7	6	simp3d 1068	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
8		breq2 4587	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝐴))
9	8	rabbidv 3164	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴})
10	9	eleq1d 2672	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)))
11	10	rspcva 3280	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))
12	1, 7, 11	syl2anc 691	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   < clt 9953   ↾_t crest 15904  SAlgcsalg 39204  SMblFncsmblfn 39586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ioo 12050  df-ico 12052  df-smblfn 39587

This theorem is referenced by:  issmfltle  39622  sssmf  39625  smfsssmf  39630  issmfle  39632  smflimlem6  39662  smfco  39687