Theorem smfpreimaltf 47508

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
smfpreimaltf.x	⊢ Ⅎ𝑥𝐹
smfpreimaltf.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpreimaltf.f	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
smfpreimaltf.d	⊢ 𝐷 = dom 𝐹
smfpreimaltf.a	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem smfpreimaltf

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpreimaltf.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		smfpreimaltf.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
3		smfpreimaltf.x	. . . . 5 ⊢ Ⅎ𝑥𝐹
4		smfpreimaltf.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
5		smfpreimaltf.d	. . . . 5 ⊢ 𝐷 = dom 𝐹
6	3, 4, 5	issmff 47506	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
7	2, 6	mpbid 235	. . 3 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
8	7	simp3d 1162	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
9		breq2 5115	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝐴))
10	9	rabbidv 3425	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴})
11	10	eleq1d 2850	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)))
12	11	rspcva 3581	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))
13	1, 8, 12	syl2anc 596	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Ⅎwnfc 2912  ∀wral 3081  {crab 3418   ⊆ wss 3906  ∪ cuni 4874   class class class wbr 5111  dom cdm 5663  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  ℝcr 11114   < clt 11258   ↾_t crest 17495  SAlgcsalg 47080  SMblFncsmblfn 47467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-pre-lttri 11189  ax-pre-lttrn 11190

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-er 8700  df-pm 8833  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-ioo 13392  df-ico 13394  df-smblfn 47468

This theorem is used by:  smfpimltmpt  47518  smfpimltxr  47519