Theorem issmflelem 40953

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 right closed 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 (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmflelem.x	|- F/ x ph
issmflelem.a	|- F/ a ph
issmflelem.s	|- ( ph -> S e. SAlg )
issmflelem.d	|- D = dom F
issmflelem.i	|- ( ph -> D C_ U. S )
issmflelem.f	|- ( ph -> F : D --> RR )
issmflelem.l	|- ( ( ph /\ a e. RR ) -> { x e. D | ( F ` x ) <_ a } e. ( S ↾t D ) )

Assertion

Ref	Expression
issmflelem	|- ( ph -> F e. (SMblFn ` S ) )

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

Allowed substitution hints:   ph( x, a)

Proof of Theorem issmflelem

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmflelem.i	. . 3 |- ( ph -> D C_ U. S )
2		issmflelem.f	. . 3 |- ( ph -> F : D --> RR )
3		issmflelem.s	. . . . . . . . . . 11 |- ( ph -> S e. SAlg )
4	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ D C_ U. S ) -> S e. SAlg )
5		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ D C_ U. S ) -> D C_ U. S )
6	4, 5	restuni4 39304	. . . . . . . . 9 |- ( ( ph /\ D C_ U. S ) -> U. ( S ↾t D ) = D )
7	6	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ D C_ U. S ) -> D = U. ( S ↾t D ) )
8	1, 7	mpdan 702	. . . . . . 7 |- ( ph -> D = U. ( S ↾t D ) )
9	8	rabeqd 39276	. . . . . 6 |- ( ph -> { x e. D | ( F ` x ) < b } = { x e. U. ( S ↾t D ) | ( F ` x ) < b } )
10	9	adantr 481	. . . . 5 |- ( ( ph /\ b e. RR ) -> { x e. D | ( F ` x ) < b } = { x e. U. ( S ↾t D ) | ( F ` x ) < b } )
11		issmflelem.x	. . . . . . 7 |- F/ x ph
12		nfv 1843	. . . . . . 7 |- F/ x b e. RR
13	11, 12	nfan 1828	. . . . . 6 |- F/ x ( ph /\ b e. RR )
14		issmflelem.a	. . . . . . 7 |- F/ a ph
15		nfv 1843	. . . . . . 7 |- F/ a b e. RR
16	14, 15	nfan 1828	. . . . . 6 |- F/ a ( ph /\ b e. RR )
17	3	uniexd 39281	. . . . . . . . . . 11 |- ( ph -> U. S e. _V )
18	17	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ D C_ U. S ) -> U. S e. _V )
19	18, 5	ssexd 4805	. . . . . . . . 9 |- ( ( ph /\ D C_ U. S ) -> D e. _V )
20		eqid 2622	. . . . . . . . 9 |- ( S ↾t D ) = ( S ↾t D )
21	4, 19, 20	subsalsal 40577	. . . . . . . 8 |- ( ( ph /\ D C_ U. S ) -> ( S ↾t D ) e. SAlg )
22	1, 21	mpdan 702	. . . . . . 7 |- ( ph -> ( S ↾t D ) e. SAlg )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ b e. RR ) -> ( S ↾t D ) e. SAlg )
24		eqid 2622	. . . . . 6 |- U. ( S ↾t D ) = U. ( S ↾t D )
25		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. U. ( S ↾t D ) ) -> x e. U. ( S ↾t D ) )
26	1, 6	mpdan 702	. . . . . . . . . . 11 |- ( ph -> U. ( S ↾t D ) = D )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. U. ( S ↾t D ) ) -> U. ( S ↾t D ) = D )
28	25, 27	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ x e. U. ( S ↾t D ) ) -> x e. D )
29	2	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> ( F ` x ) e. RR )
30	28, 29	syldan 487	. . . . . . . 8 |- ( ( ph /\ x e. U. ( S ↾t D ) ) -> ( F ` x ) e. RR )
31	30	rexrd 10089	. . . . . . 7 |- ( ( ph /\ x e. U. ( S ↾t D ) ) -> ( F ` x ) e. RR* )
32	31	adantlr 751	. . . . . 6 |- ( ( ( ph /\ b e. RR ) /\ x e. U. ( S ↾t D ) ) -> ( F ` x ) e. RR* )
33	26	rabeqd 39276	. . . . . . . . 9 |- ( ph -> { x e. U. ( S ↾t D ) | ( F ` x ) <_ a } = { x e. D | ( F ` x ) <_ a } )
34	33	adantr 481	. . . . . . . 8 |- ( ( ph /\ a e. RR ) -> { x e. U. ( S ↾t D ) | ( F ` x ) <_ a } = { x e. D | ( F ` x ) <_ a } )
35		issmflelem.l	. . . . . . . 8 |- ( ( ph /\ a e. RR ) -> { x e. D | ( F ` x ) <_ a } e. ( S ↾t D ) )
36	34, 35	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> { x e. U. ( S ↾t D ) | ( F ` x ) <_ a } e. ( S ↾t D ) )
37	36	adantlr 751	. . . . . 6 |- ( ( ( ph /\ b e. RR ) /\ a e. RR ) -> { x e. U. ( S ↾t D ) | ( F ` x ) <_ a } e. ( S ↾t D ) )
38		simpr 477	. . . . . 6 |- ( ( ph /\ b e. RR ) -> b e. RR )
39	13, 16, 23, 24, 32, 37, 38	salpreimalelt 40938	. . . . 5 |- ( ( ph /\ b e. RR ) -> { x e. U. ( S ↾t D ) | ( F ` x ) < b } e. ( S ↾t D ) )
40	10, 39	eqeltrd 2701	. . . 4 |- ( ( ph /\ b e. RR ) -> { x e. D | ( F ` x ) < b } e. ( S ↾t D ) )
41	40	ralrimiva 2966	. . 3 |- ( ph -> A. b e. RR { x e. D | ( F ` x ) < b } e. ( S ↾t D ) )
42	1, 2, 41	3jca 1242	. 2 |- ( ph -> ( D C_ U. S /\ F : D --> RR /\ A. b e. RR { x e. D | ( F ` x ) < b } e. ( S ↾t D ) ) )
43		issmflelem.d	. . 3 |- D = dom F
44	3, 43	issmf 40937	. 2 |- ( ph -> ( F e. (SMblFn ` S ) <-> ( D C_ U. S /\ F : D --> RR /\ A. b e. RR { x e. D | ( F ` x ) < b } e. ( S ↾t D ) ) ) )
45	42, 44	mpbird 247	1 |- ( ph -> F e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   ↾_t crest 16081  SAlgcsalg 40528  SMblFncsmblfn 40909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  issmfle  40954