Theorem smfres 40997

Description: The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfres.s	|- ( ph -> S e. SAlg )
smfres.f	|- ( ph -> F e. (SMblFn ` S ) )
smfres.a	|- ( ph -> A e. V )

Assertion

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

Proof of Theorem smfres

Dummy variables x a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ a ph
2		smfres.s	. 2 |- ( ph -> S e. SAlg )
3		inss1 3833	. . . 4 |- ( dom F i^i A ) C_ dom F
4	3	a1i 11	. . 3 |- ( ph -> ( dom F i^i A ) C_ dom F )
5		smfres.f	. . . 4 |- ( ph -> F e. (SMblFn ` S ) )
6		eqid 2622	. . . 4 |- dom F = dom F
7	2, 5, 6	smfdmss 40942	. . 3 |- ( ph -> dom F C_ U. S )
8	4, 7	sstrd 3613	. 2 |- ( ph -> ( dom F i^i A ) C_ U. S )
9	2, 5, 6	smff 40941	. . 3 |- ( ph -> F : dom F --> RR )
10		fresin 6073	. . 3 |- ( F : dom F --> RR -> ( F |` A ) : ( dom F i^i A ) --> RR )
11	9, 10	syl 17	. 2 |- ( ph -> ( F |` A ) : ( dom F i^i A ) --> RR )
12		ovexd 6680	. . . 4 |- ( ( ph /\ a e. RR ) -> ( S ↾t dom F ) e. _V )
13		smfres.a	. . . . 5 |- ( ph -> A e. V )
14	13	adantr 481	. . . 4 |- ( ( ph /\ a e. RR ) -> A e. V )
15	2	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> S e. SAlg )
16	5	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> F e. (SMblFn ` S ) )
17		mnfxr 10096	. . . . . 6 |- -oo e. RR*
18	17	a1i 11	. . . . 5 |- ( ( ph /\ a e. RR ) -> -oo e. RR* )
19		rexr 10085	. . . . . 6 |- ( a e. RR -> a e. RR* )
20	19	adantl 482	. . . . 5 |- ( ( ph /\ a e. RR ) -> a e. RR* )
21	15, 16, 6, 18, 20	smfpimioo 40994	. . . 4 |- ( ( ph /\ a e. RR ) -> ( `' F " ( -oo (,) a ) ) e. ( S ↾t dom F ) )
22		eqid 2622	. . . 4 |- ( ( `' F " ( -oo (,) a ) ) i^i A ) = ( ( `' F " ( -oo (,) a ) ) i^i A )
23	12, 14, 21, 22	elrestd 39291	. . 3 |- ( ( ph /\ a e. RR ) -> ( ( `' F " ( -oo (,) a ) ) i^i A ) e. ( ( S ↾t dom F ) ↾t A ) )
24	9	ffund 6049	. . . . . . . 8 |- ( ph -> Fun F )
25		respreima 6344	. . . . . . . 8 |- ( Fun F -> ( `' ( F |` A ) " ( -oo (,) a ) ) = ( ( `' F " ( -oo (,) a ) ) i^i A ) )
26	24, 25	syl 17	. . . . . . 7 |- ( ph -> ( `' ( F |` A ) " ( -oo (,) a ) ) = ( ( `' F " ( -oo (,) a ) ) i^i A ) )
27	26	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( `' F " ( -oo (,) a ) ) i^i A ) = ( `' ( F |` A ) " ( -oo (,) a ) ) )
28	27	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( ( `' F " ( -oo (,) a ) ) i^i A ) = ( `' ( F |` A ) " ( -oo (,) a ) ) )
29	11	adantr 481	. . . . . 6 |- ( ( ph /\ a e. RR ) -> ( F |` A ) : ( dom F i^i A ) --> RR )
30	29, 20	preimaioomnf 40929	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( `' ( F |` A ) " ( -oo (,) a ) ) = { x e. ( dom F i^i A ) | ( ( F |` A ) ` x ) < a } )
31	28, 30	eqtr2d 2657	. . . 4 |- ( ( ph /\ a e. RR ) -> { x e. ( dom F i^i A ) | ( ( F |` A ) ` x ) < a } = ( ( `' F " ( -oo (,) a ) ) i^i A ) )
32	5	dmexd 39422	. . . . . . 7 |- ( ph -> dom F e. _V )
33		restco 20968	. . . . . . 7 |- ( ( S e. SAlg /\ dom F e. _V /\ A e. V ) -> ( ( S ↾t dom F ) ↾t A ) = ( S ↾t ( dom F i^i A ) ) )
34	2, 32, 13, 33	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( S ↾t dom F ) ↾t A ) = ( S ↾t ( dom F i^i A ) ) )
35	34	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( ( S ↾t dom F ) ↾t A ) = ( S ↾t ( dom F i^i A ) ) )
36	35	eqcomd 2628	. . . 4 |- ( ( ph /\ a e. RR ) -> ( S ↾t ( dom F i^i A ) ) = ( ( S ↾t dom F ) ↾t A ) )
37	31, 36	eleq12d 2695	. . 3 |- ( ( ph /\ a e. RR ) -> ( { x e. ( dom F i^i A ) | ( ( F |` A ) ` x ) < a } e. ( S ↾t ( dom F i^i A ) ) <-> ( ( `' F " ( -oo (,) a ) ) i^i A ) e. ( ( S ↾t dom F ) ↾t A ) ) )
38	23, 37	mpbird 247	. 2 |- ( ( ph /\ a e. RR ) -> { x e. ( dom F i^i A ) | ( ( F |` A ) ` x ) < a } e. ( S ↾t ( dom F i^i A ) ) )
39	1, 2, 8, 11, 38	issmfd 40944	1 |- ( ph -> ( F |` A ) e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   (,)cioo 12175   ↾_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: (None)