Theorem issmff 47014

Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmff.x	⊢ Ⅎ𝑥𝐹
issmff.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmff.d	⊢ 𝐷 = dom 𝐹

Assertion

Ref	Expression
issmff	⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))

Distinct variable groups:   𝐷,𝑎   𝐹,𝑎   𝑆,𝑎   𝑥,𝑎

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

Proof of Theorem issmff

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmff.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2		issmff.d	. . 3 ⊢ 𝐷 = dom 𝐹
3	1, 2	issmf 47008	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
4		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦𝐷
5		issmff.x	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6	5	nfdm 5901	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝐹
7	2, 6	nfcxfr 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
8		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
9	5, 8	nffv 6845	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
10		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥 <
11		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
12	9, 10, 11	nfbr 5146	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) < 𝑎
13		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥) < 𝑎
14		fveq2 6835	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
15	14	breq1d 5109	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
16	4, 7, 12, 13, 15	cbvrabw 3435	. . . . . 6 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}
17	16	eleq1i 2828	. . . . 5 ⊢ ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
18	17	ralbii 3083	. . . 4 ⊢ (∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
19	18	3anbi3i 1160	. . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
20	19	a1i 11	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
21	3, 20	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  {crab 3400   ⊆ wss 3902  ∪ cuni 4864   class class class wbr 5099  dom cdm 5625  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  ℝcr 11029   < clt 11170   ↾_t crest 17344  SAlgcsalg 46588  SMblFncsmblfn 46975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-pre-lttri 11104  ax-pre-lttrn 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-er 8637  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-ioo 13269  df-ico 13271  df-smblfn 46976

This theorem is referenced by:  smfpreimaltf  47016  issmfdf  47017