Theorem smfpimltmpt 41475

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 4881	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		smfpimltmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		smfpimltmpt.f	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
4		eqid 2771	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
5		smfpimltmpt.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
6	1, 2, 3, 4, 5	smfpreimaltf 41465	. 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 2771	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimltmpt.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
10	7, 8, 9	dmmptdf 39935	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	1	nfdm 5505	. . . . . 6 ⊢ Ⅎ𝑥dom (𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfcv 2913	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	rabeqf 3340	. . . . 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 17	. . . 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 6435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	breq1d 4796	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅 ↔ 𝐵 < 𝑅))
18	7, 17	rabbida 39795	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
19		eqidd 2772	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
20	14, 18, 19	3eqtrrd 2810	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅})
21	10	eqcomd 2777	. . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	oveq2d 6809	. . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) = (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))
23	20, 22	eleq12d 2844	. 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 247	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631  Ⅎwnf 1856   ∈ wcel 2145  {crab 3065   class class class wbr 4786   ↦ cmpt 4863  dom cdm 5249  ‘cfv 6031  (class class class)co 6793  ℝcr 10137   < clt 10276   ↾_t crest 16289  SAlgcsalg 41045  SMblFncsmblfn 41429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-ioo 12384  df-ico 12386  df-smblfn 41430

This theorem is referenced by:  smfaddlem2  41492  smfrec  41516