Theorem smfpimltmpt 46986

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 5197	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		smfpimltmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		smfpimltmpt.f	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
4		eqid 2736	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
5		smfpimltmpt.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
6	1, 2, 3, 4, 5	smfpreimaltf 46976	. 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 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimltmpt.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
10	7, 8, 9	dmmptdf 45464	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	1	nfdm 5900	. . . . . 6 ⊢ Ⅎ𝑥dom (𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	rabeqf 3433	. . . . 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 6954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	breq1d 5108	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅 ↔ 𝐵 < 𝑅))
18	7, 17	rabbida 3425	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
19		eqidd 2737	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅})
20	14, 18, 19	3eqtrrd 2776	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} = {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑅})
21	10	eqcomd 2742	. . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	oveq2d 7374	. . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) = (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))
23	20, 22	eleq12d 2830	. 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 257	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  {crab 3399   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  ℝcr 11025   < clt 11166   ↾_t crest 17340  SAlgcsalg 46548  SMblFncsmblfn 46935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-er 8635  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-ioo 13265  df-ico 13267  df-smblfn 46936

This theorem is referenced by:  smfaddlem2  47004  smfrec  47029