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Theorem issmfge 43844
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 484 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 488 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfge.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 43808 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 43807 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1921 . . . . . . . . 9 𝑦𝜑
8 nfv 1921 . . . . . . . . 9 𝑦 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1906 . . . . . . . 8 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1921 . . . . . . . 8 𝑦 𝑏 ∈ ℝ
119, 10nfan 1906 . . . . . . 7 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12 nfv 1921 . . . . . . 7 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
131uniexd 7486 . . . . . . . . . . . 12 (𝜑 𝑆 ∈ V)
1413adantr 484 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
15 simpr 488 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1614, 15ssexd 5192 . . . . . . . . . 10 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
175, 16syldan 594 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18 eqid 2738 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
192, 17, 18subsalsal 43440 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2019adantr 484 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
216ffvelrnda 6861 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ)
2221rexrd 10769 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
2322adantlr 715 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
242adantr 484 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
253adantr 484 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
26 simpr 488 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
2724, 25, 4, 26smfpreimagt 43836 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
2827adantlr 715 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
29 simpr 488 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3011, 12, 20, 23, 28, 29salpreimagtge 43800 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
3130ralrimiva 3096 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
325, 6, 313jca 1129 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
3332ex 416 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
34 nfv 1921 . . . . . . 7 𝑦 𝐷 𝑆
35 nfv 1921 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
36 nfcv 2899 . . . . . . . 8 𝑦
37 nfrab1 3287 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)}
38 nfcv 2899 . . . . . . . . 9 𝑦(𝑆t 𝐷)
3937, 38nfel 2913 . . . . . . . 8 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4036, 39nfralw 3138 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4134, 35, 40nf3an 1908 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
427, 41nfan 1906 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
43 nfv 1921 . . . . . 6 𝑏𝜑
44 nfv 1921 . . . . . . 7 𝑏 𝐷 𝑆
45 nfv 1921 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
46 nfra1 3131 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4744, 45, 46nf3an 1908 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
4843, 47nfan 1906 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
491adantr 484 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
50 simpr1 1195 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
51 simpr2 1196 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
52 simpr3 1197 . . . . 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 43843 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5453ex 416 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
5533, 54impbid 215 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
56 breq1 5033 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑦)))
5756rabbidv 3381 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑦𝐷𝑎 ≤ (𝐹𝑦)})
58 fveq2 6674 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
5958breq2d 5042 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑥)))
6059cbvrabv 3393 . . . . . . . 8 {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)}
6160a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6257, 61eqtrd 2773 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6362eleq1d 2817 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6463cbvralvw 3349 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))
65643anbi3i 1160 . . 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 282 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  {crab 3057  Vcvv 3398  wss 3843   cuni 4796   class class class wbr 5030  dom cdm 5525  wf 6335  cfv 6339  (class class class)co 7170  cr 10614  *cxr 10752   < clt 10753  cle 10754  t crest 16797  SAlgcsalg 43391  SMblFncsmblfn 43775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cc 9935  ax-ac2 9963  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-card 9441  df-acn 9444  df-ac 9616  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-q 12431  df-rp 12473  df-ioo 12825  df-ico 12827  df-fl 13253  df-rest 16799  df-salg 43392  df-smblfn 43776
This theorem is referenced by:  smfpreimage  43856
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