Theorem smfpreimalt 47682

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 47679	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
6	2, 5	mpbid 235	. . 3 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
7	6	simp3d 1162	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
8		breq2 5107	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝐴))
9	8	rabbidv 3420	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴})
10	9	eleq1d 2846	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)))
11	10	rspcva 3575	. 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 596	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  dom cdm 5651  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  ℝcr 11177   < clt 11321   ↾_t crest 17568  SAlgcsalg 47259  SMblFncsmblfn 47646

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11234  ax-resscn 11235  ax-pre-lttri 11252  ax-pre-lttrn 11253

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-er 8701  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11323  df-mnf 11324  df-xr 11325  df-ltxr 11326  df-le 11327  df-ioo 13458  df-ico 13460  df-smblfn 47647

This theorem is used by:  sssmf  47689  smfsssmf  47694  issmfle  47696  smflimlem6  47727  smfco  47753