Theorem smfpimltmpt 47443

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
smfpimltmpt.x	⊢ Ⅎ𝑥𝜑
smfpimltmpt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpimltmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
smfpimltmpt.f	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smfpimltmpt.r	⊢ (𝜑 → 𝑅 ∈ ℝ)

Assertion

Ref	Expression
smfpimltmpt	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

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

Proof of Theorem smfpimltmpt

Step	Hyp	Ref	Expression
1		nfmpt1 5211	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		smfpimltmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		smfpimltmpt.f	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
4		eqid 2763	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
5		smfpimltmpt.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
6	1, 2, 3, 4, 5	smfpreimaltf 47433	. 2 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))
7		smfpimltmpt.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
8		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimltmpt.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
10	7, 8, 9	dmmptdf 45923	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	1	nfdm 5943	. . . . . 6 ⊢ Ⅎ𝑥dom (𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	rabeqf 3450	. . . . 5 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} = {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅})
14	10, 13	syl 18	. . . 4 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} = {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅})
15	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
16	15, 9	fvmpt2d 7005	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	breq1d 5120	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅 ↔ 𝐵 < 𝑅))
18	7, 17	rabbida 3442	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
19		eqidd 2764	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
20	14, 18, 19	3eqtrrd 2803	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅})
21	10	eqcomd 2769	. . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	oveq2d 7428	. . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) = (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))
23	20, 22	eleq12d 2857	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴) ↔ {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))))
24	6, 23	mpbird 260	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  {crab 3416   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  ℝcr 11100   < clt 11244   ↾_t crest 17474  SAlgcsalg 47005  SMblFncsmblfn 47392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-er 8695  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-ioo 13377  df-ico 13379  df-smblfn 47393

This theorem is referenced by:  smfaddlem2  47461  smfrec  47486