Theorem sssmf 40947

Description: The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
sssmf.s	|- ( ph -> S e. SAlg )
sssmf.f	|- ( ph -> F e. (SMblFn ` S ) )

Assertion

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

Proof of Theorem sssmf

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

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ a ph
2		sssmf.s	. 2 |- ( ph -> S e. SAlg )
3		inss2 3834	. . 3 |- ( B i^i dom F ) C_ dom F
4		sssmf.f	. . . 4 |- ( ph -> F e. (SMblFn ` S ) )
5		eqid 2622	. . . 4 |- dom F = dom F
6	2, 4, 5	smfdmss 40942	. . 3 |- ( ph -> dom F C_ U. S )
7	3, 6	syl5ss 3614	. 2 |- ( ph -> ( B i^i dom F ) C_ U. S )
8	2, 4, 5	smff 40941	. . . . 5 |- ( ph -> F : dom F --> RR )
9	3	a1i 11	. . . . 5 |- ( ph -> ( B i^i dom F ) C_ dom F )
10		fssres 6070	. . . . 5 |- ( ( F : dom F --> RR /\ ( B i^i dom F ) C_ dom F ) -> ( F |` ( B i^i dom F ) ) : ( B i^i dom F ) --> RR )
11	8, 9, 10	syl2anc 693	. . . 4 |- ( ph -> ( F |` ( B i^i dom F ) ) : ( B i^i dom F ) --> RR )
12	8	freld 39425	. . . . . . 7 |- ( ph -> Rel F )
13		resindm 5444	. . . . . . 7 |- ( Rel F -> ( F |` ( B i^i dom F ) ) = ( F |` B ) )
14	12, 13	syl 17	. . . . . 6 |- ( ph -> ( F |` ( B i^i dom F ) ) = ( F |` B ) )
15	14	eqcomd 2628	. . . . 5 |- ( ph -> ( F |` B ) = ( F |` ( B i^i dom F ) ) )
16		dmres 5419	. . . . . 6 |- dom ( F |` B ) = ( B i^i dom F )
17	16	a1i 11	. . . . 5 |- ( ph -> dom ( F |` B ) = ( B i^i dom F ) )
18	15, 17	feq12d 6033	. . . 4 |- ( ph -> ( ( F |` B ) : dom ( F |` B ) --> RR <-> ( F |` ( B i^i dom F ) ) : ( B i^i dom F ) --> RR ) )
19	11, 18	mpbird 247	. . 3 |- ( ph -> ( F |` B ) : dom ( F |` B ) --> RR )
20	17	feq2d 6031	. . 3 |- ( ph -> ( ( F |` B ) : dom ( F |` B ) --> RR <-> ( F |` B ) : ( B i^i dom F ) --> RR ) )
21	19, 20	mpbid 222	. 2 |- ( ph -> ( F |` B ) : ( B i^i dom F ) --> RR )
22	2	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> S e. SAlg )
23	4	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> F e. (SMblFn ` S ) )
24		simpr 477	. . . . 5 |- ( ( ph /\ a e. RR ) -> a e. RR )
25	22, 23, 5, 24	smfpreimalt 40940	. . . 4 |- ( ( ph /\ a e. RR ) -> { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) )
26	4	dmexd 39422	. . . . . 6 |- ( ph -> dom F e. _V )
27		elrest 16088	. . . . . 6 |- ( ( S e. SAlg /\ dom F e. _V ) -> ( { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) <-> E. w e. S { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) )
28	2, 26, 27	syl2anc 693	. . . . 5 |- ( ph -> ( { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) <-> E. w e. S { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) )
29	28	adantr 481	. . . 4 |- ( ( ph /\ a e. RR ) -> ( { x e. dom F | ( F ` x ) < a } e. ( S ↾t dom F ) <-> E. w e. S { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) )
30	25, 29	mpbid 222	. . 3 |- ( ( ph /\ a e. RR ) -> E. w e. S { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) )
31		elinel1 3799	. . . . . . . . . . . . 13 |- ( x e. ( B i^i dom F ) -> x e. B )
32	31	fvresd 6208	. . . . . . . . . . . 12 |- ( x e. ( B i^i dom F ) -> ( ( F |` B ) ` x ) = ( F ` x ) )
33	32	breq1d 4663	. . . . . . . . . . 11 |- ( x e. ( B i^i dom F ) -> ( ( ( F |` B ) ` x ) < a <-> ( F ` x ) < a ) )
34	33	rabbiia 3185	. . . . . . . . . 10 |- { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } = { x e. ( B i^i dom F ) | ( F ` x ) < a }
35	34	a1i 11	. . . . . . . . 9 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } = { x e. ( B i^i dom F ) | ( F ` x ) < a } )
36		rabss2 3685	. . . . . . . . . . . . 13 |- ( ( B i^i dom F ) C_ dom F -> { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ { x e. dom F | ( F ` x ) < a } )
37	3, 36	ax-mp 5	. . . . . . . . . . . 12 |- { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ { x e. dom F | ( F ` x ) < a }
38		id 22	. . . . . . . . . . . . 13 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) )
39		inss1 3833	. . . . . . . . . . . . . 14 |- ( w i^i dom F ) C_ w
40	39	a1i 11	. . . . . . . . . . . . 13 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( w i^i dom F ) C_ w )
41	38, 40	eqsstrd 3639	. . . . . . . . . . . 12 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. dom F | ( F ` x ) < a } C_ w )
42	37, 41	syl5ss 3614	. . . . . . . . . . 11 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ w )
43		ssrab2 3687	. . . . . . . . . . . 12 |- { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ ( B i^i dom F )
44	43	a1i 11	. . . . . . . . . . 11 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ ( B i^i dom F ) )
45	42, 44	ssind 3837	. . . . . . . . . 10 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( F ` x ) < a } C_ ( w i^i ( B i^i dom F ) ) )
46		nfrab1 3122	. . . . . . . . . . . . 13 |- F/_ x { x e. dom F | ( F ` x ) < a }
47		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x ( w i^i dom F )
48	46, 47	nfeq 2776	. . . . . . . . . . . 12 |- F/ x { x e. dom F | ( F ` x ) < a } = ( w i^i dom F )
49		elinel2 3800	. . . . . . . . . . . . . . . . . . 19 |- ( x e. ( w i^i ( B i^i dom F ) ) -> x e. ( B i^i dom F ) )
50	49	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> x e. ( B i^i dom F ) )
51		elinel1 3799	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x e. ( w i^i ( B i^i dom F ) ) -> x e. w )
52	3	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( x e. ( B i^i dom F ) -> x e. dom F )
53	49, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x e. ( w i^i ( B i^i dom F ) ) -> x e. dom F )
54	51, 53	elind 3798	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( w i^i ( B i^i dom F ) ) -> x e. ( w i^i dom F ) )
55	54	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> x e. ( w i^i dom F ) )
56	38	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( w i^i dom F ) = { x e. dom F | ( F ` x ) < a } )
57	56	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> ( w i^i dom F ) = { x e. dom F | ( F ` x ) < a } )
58	55, 57	eleqtrd 2703	. . . . . . . . . . . . . . . . . . 19 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> x e. { x e. dom F | ( F ` x ) < a } )
59		rabid 3116	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. { x e. dom F | ( F ` x ) < a } <-> ( x e. dom F /\ ( F ` x ) < a ) )
60	59	biimpi 206	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. { x e. dom F | ( F ` x ) < a } -> ( x e. dom F /\ ( F ` x ) < a ) )
61	60	simprd 479	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { x e. dom F | ( F ` x ) < a } -> ( F ` x ) < a )
62	58, 61	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> ( F ` x ) < a )
63	50, 62	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> ( x e. ( B i^i dom F ) /\ ( F ` x ) < a ) )
64		rabid 3116	. . . . . . . . . . . . . . . . 17 |- ( x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } <-> ( x e. ( B i^i dom F ) /\ ( F ` x ) < a ) )
65	63, 64	sylibr 224	. . . . . . . . . . . . . . . 16 |- ( ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) /\ x e. ( w i^i ( B i^i dom F ) ) ) -> x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } )
66	65	ex 450	. . . . . . . . . . . . . . 15 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( x e. ( w i^i ( B i^i dom F ) ) -> x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } ) )
67	48, 66	ralrimi 2957	. . . . . . . . . . . . . 14 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> A. x e. ( w i^i ( B i^i dom F ) ) x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } )
68		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ x ( w i^i ( B i^i dom F ) )
69		nfrab1 3122	. . . . . . . . . . . . . . 15 |- F/_ x { x e. ( B i^i dom F ) | ( F ` x ) < a }
70	68, 69	dfss3f 3595	. . . . . . . . . . . . . 14 |- ( ( w i^i ( B i^i dom F ) ) C_ { x e. ( B i^i dom F ) | ( F ` x ) < a } <-> A. x e. ( w i^i ( B i^i dom F ) ) x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } )
71	67, 70	sylibr 224	. . . . . . . . . . . . 13 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( w i^i ( B i^i dom F ) ) C_ { x e. ( B i^i dom F ) | ( F ` x ) < a } )
72	71	sseld 3602	. . . . . . . . . . . 12 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( x e. ( w i^i ( B i^i dom F ) ) -> x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } ) )
73	48, 72	ralrimi 2957	. . . . . . . . . . 11 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> A. x e. ( w i^i ( B i^i dom F ) ) x e. { x e. ( B i^i dom F ) | ( F ` x ) < a } )
74	73, 70	sylibr 224	. . . . . . . . . 10 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> ( w i^i ( B i^i dom F ) ) C_ { x e. ( B i^i dom F ) | ( F ` x ) < a } )
75	45, 74	eqssd 3620	. . . . . . . . 9 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( F ` x ) < a } = ( w i^i ( B i^i dom F ) ) )
76	35, 75	eqtrd 2656	. . . . . . . 8 |- ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } = ( w i^i ( B i^i dom F ) ) )
77	76	adantl 482	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } = ( w i^i ( B i^i dom F ) ) )
78	77	3adant2 1080	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } = ( w i^i ( B i^i dom F ) ) )
79	22	3ad2ant1 1082	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> S e. SAlg )
80		simp1l 1085	. . . . . . . 8 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> ph )
81	26, 9	ssexd 4805	. . . . . . . 8 |- ( ph -> ( B i^i dom F ) e. _V )
82	80, 81	syl 17	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> ( B i^i dom F ) e. _V )
83		simp2 1062	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> w e. S )
84		eqid 2622	. . . . . . 7 |- ( w i^i ( B i^i dom F ) ) = ( w i^i ( B i^i dom F ) )
85	79, 82, 83, 84	elrestd 39291	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> ( w i^i ( B i^i dom F ) ) e. ( S ↾t ( B i^i dom F ) ) )
86	78, 85	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ w e. S /\ { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } e. ( S ↾t ( B i^i dom F ) ) )
87	86	3exp 1264	. . . 4 |- ( ( ph /\ a e. RR ) -> ( w e. S -> ( { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } e. ( S ↾t ( B i^i dom F ) ) ) ) )
88	87	rexlimdv 3030	. . 3 |- ( ( ph /\ a e. RR ) -> ( E. w e. S { x e. dom F | ( F ` x ) < a } = ( w i^i dom F ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } e. ( S ↾t ( B i^i dom F ) ) ) )
89	30, 88	mpd 15	. 2 |- ( ( ph /\ a e. RR ) -> { x e. ( B i^i dom F ) | ( ( F |` B ) ` x ) < a } e. ( S ↾t ( B i^i dom F ) ) )
90	1, 2, 7, 21, 89	issmfd 40944	1 |- ( ph -> ( F |` B ) e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Rel wrel 5119   -->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-rep 4771  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-reu 2919  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-rest 16083  df-smblfn 40910

This theorem is referenced by:  sssmfmpt  40959