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Theorem issmfge 44305
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 left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmfge.s (𝜑𝑆 ∈ SAlg)
issmfge.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmfge (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfge
Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmfge.s . . . . . . 7 (𝜑𝑆 ∈ SAlg)
21adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 485 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfge.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 44269 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 44268 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1917 . . . . . . . . 9 𝑦𝜑
8 nfv 1917 . . . . . . . . 9 𝑦 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1902 . . . . . . . 8 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1917 . . . . . . . 8 𝑦 𝑏 ∈ ℝ
119, 10nfan 1902 . . . . . . 7 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12 nfv 1917 . . . . . . 7 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
131uniexd 7595 . . . . . . . . . . . 12 (𝜑 𝑆 ∈ V)
1413adantr 481 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
15 simpr 485 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1614, 15ssexd 5248 . . . . . . . . . 10 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
175, 16syldan 591 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18 eqid 2738 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
192, 17, 18subsalsal 43898 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2019adantr 481 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
216ffvelrnda 6961 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ)
2221rexrd 11025 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
2322adantlr 712 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
242adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
253adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
26 simpr 485 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
2724, 25, 4, 26smfpreimagt 44297 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
2827adantlr 712 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
29 simpr 485 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3011, 12, 20, 23, 28, 29salpreimagtge 44261 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
3130ralrimiva 3103 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
325, 6, 313jca 1127 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
3332ex 413 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
34 nfv 1917 . . . . . . 7 𝑦 𝐷 𝑆
35 nfv 1917 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
36 nfcv 2907 . . . . . . . 8 𝑦
37 nfrab1 3317 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)}
38 nfcv 2907 . . . . . . . . 9 𝑦(𝑆t 𝐷)
3937, 38nfel 2921 . . . . . . . 8 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4036, 39nfralw 3151 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4134, 35, 40nf3an 1904 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
427, 41nfan 1902 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
43 nfv 1917 . . . . . 6 𝑏𝜑
44 nfv 1917 . . . . . . 7 𝑏 𝐷 𝑆
45 nfv 1917 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
46 nfra1 3144 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4744, 45, 46nf3an 1904 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
4843, 47nfan 1902 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
491adantr 481 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
50 simpr1 1193 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
51 simpr2 1194 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
52 simpr3 1195 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
5342, 48, 49, 4, 50, 51, 52issmfgelem 44304 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5453ex 413 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
5533, 54impbid 211 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
56 breq1 5077 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑦)))
5756rabbidv 3414 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑦𝐷𝑎 ≤ (𝐹𝑦)})
58 fveq2 6774 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
5958breq2d 5086 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑥)))
6059cbvrabv 3426 . . . . . . . 8 {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)}
6160a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6257, 61eqtrd 2778 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6362eleq1d 2823 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6463cbvralvw 3383 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))
65643anbi3i 1158 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6665a1i 11 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
6755, 66bitrd 278 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3432  wss 3887   cuni 4839   class class class wbr 5074  dom cdm 5589  wf 6429  cfv 6433  (class class class)co 7275  cr 10870  *cxr 11008   < clt 11009  cle 11010  t crest 17131  SAlgcsalg 43849  SMblFncsmblfn 44233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-ioo 13083  df-ico 13085  df-fl 13512  df-rest 17133  df-salg 43850  df-smblfn 44234
This theorem is referenced by:  smfpreimage  44317
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