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Theorem issmfge 40301
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 477 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfge.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 40265 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 40264 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1840 . . . . . . . . 9 𝑦𝜑
8 nfv 1840 . . . . . . . . 9 𝑦 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1825 . . . . . . . 8 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1840 . . . . . . . 8 𝑦 𝑏 ∈ ℝ
119, 10nfan 1825 . . . . . . 7 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12 nfv 1840 . . . . . . 7 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
131uniexd 38785 . . . . . . . . . . . 12 (𝜑 𝑆 ∈ V)
1413adantr 481 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
15 simpr 477 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1614, 15ssexd 4767 . . . . . . . . . 10 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
175, 16syldan 487 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18 eqid 2621 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
192, 17, 18subsalsal 39900 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2019adantr 481 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
216ffvelrnda 6317 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ)
2221rexrd 10036 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
2322adantlr 750 . . . . . . 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 477 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
2724, 25, 4, 26smfpreimagt 40293 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
2827adantlr 750 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
29 simpr 477 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3011, 12, 20, 23, 28, 29salpreimagtge 40257 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
3130ralrimiva 2960 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
325, 6, 313jca 1240 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
3332ex 450 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
34 nfv 1840 . . . . . . 7 𝑦 𝐷 𝑆
35 nfv 1840 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
36 nfcv 2761 . . . . . . . 8 𝑦
37 nfrab1 3111 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)}
38 nfcv 2761 . . . . . . . . 9 𝑦(𝑆t 𝐷)
3937, 38nfel 2773 . . . . . . . 8 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4036, 39nfral 2940 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4134, 35, 40nf3an 1828 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
427, 41nfan 1825 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
43 nfv 1840 . . . . . 6 𝑏𝜑
44 nfv 1840 . . . . . . 7 𝑏 𝐷 𝑆
45 nfv 1840 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
46 nfra1 2936 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4744, 45, 46nf3an 1828 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
4843, 47nfan 1825 . . . . 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 1065 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
51 simpr2 1066 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
52 simpr3 1067 . . . . 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 40300 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5453ex 450 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
5533, 54impbid 202 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
56 breq1 4618 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑦)))
5756rabbidv 3177 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑦𝐷𝑎 ≤ (𝐹𝑦)})
58 fveq2 6150 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
5958breq2d 4627 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑥)))
6059cbvrabv 3185 . . . . . . . 8 {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)}
6160a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6257, 61eqtrd 2655 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6362eleq1d 2683 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6463cbvralv 3159 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))
65643anbi3i 1253 . . 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 268 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3556   cuni 4404   class class class wbr 4615  dom cdm 5076  wf 5845  cfv 5849  (class class class)co 6607  cr 9882  *cxr 10020   < clt 10021  cle 10022  t crest 16005  SAlgcsalg 39851  SMblFncsmblfn 40232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cc 9204  ax-ac2 9232  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-sup 8295  df-inf 8296  df-card 8712  df-acn 8715  df-ac 8886  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-n0 11240  df-z 11325  df-uz 11635  df-q 11736  df-rp 11780  df-ioo 12124  df-ico 12126  df-fl 12536  df-rest 16007  df-salg 39852  df-smblfn 40233
This theorem is referenced by:  smfpreimage  40313
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