Theorem issmfdf 40946

Description: A sufficient condition for " F being a measurable function w.r.t. to the sigma-algebra S". (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem issmfdf

Step	Hyp	Ref	Expression
1		issmfdf.f	. . . . 5 |- ( ph -> F : D --> RR )
2	1	fdmd 39420	. . . 4 |- ( ph -> dom F = D )
3		issmfdf.d	. . . 4 |- ( ph -> D C_ U. S )
4	2, 3	eqsstrd 3639	. . 3 |- ( ph -> dom F C_ U. S )
5	1	ffdmd 6063	. . 3 |- ( ph -> F : dom F --> RR )
6		issmfdf.a	. . . 4 |- F/ a ph
7		issmfdf.p	. . . . . 6 |- ( ( ph /\ a e. RR ) -> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) )
8		issmfdf.x	. . . . . . . . . . 11 |- F/_ x F
9	8	nfdm 5367	. . . . . . . . . 10 |- F/_ x dom F
10		nfcv 2764	. . . . . . . . . 10 |- F/_ x D
11	9, 10	rabeqf 3190	. . . . . . . . 9 |- ( dom F = D -> { x e. dom F | ( F ` x ) < a } = { x e. D | ( F ` x ) < a } )
12	2, 11	syl 17	. . . . . . . 8 |- ( ph -> { x e. dom F | ( F ` x ) < a } = { x e. D | ( F ` x ) < a } )
13	2	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( S ↾t dom F ) = ( S ↾t D ) )
14	12, 13	eleq12d 2695	. . . . . . 7 |- ( ph -> ( { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) <-> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
15	14	adantr 481	. . . . . 6 |- ( ( ph /\ a e. RR ) -> ( { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) <-> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
16	7, 15	mpbird 247	. . . . 5 |- ( ( ph /\ a e. RR ) -> { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) )
17	16	ex 450	. . . 4 |- ( ph -> ( a e. RR -> { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) ) )
18	6, 17	ralrimi 2957	. . 3 |- ( ph -> A. a e. RR { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) )
19	4, 5, 18	3jca 1242	. 2 |- ( ph -> ( dom F C_ U. S /\ F : dom F --> RR /\ A. a e. RR { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) ) )
20		issmfdf.s	. . 3 |- ( ph -> S e. SAlg )
21		eqid 2622	. . 3 |- dom F = dom F
22	8, 20, 21	issmff 40943	. 2 |- ( ph -> ( F e. (SMblFn ` S ) <-> ( dom F C_ U. S /\ F : dom F --> RR /\ A. a e. RR { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) ) ) )
23	19, 22	mpbird 247	1 |- ( ph -> F e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   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   < clt 10074   ↾_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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179  df-ico 12181  df-smblfn 40910

This theorem is referenced by:  issmfdmpt  40957  smfconst  40958