Theorem ismbf 23397

Description: The predicate " F is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 23294. (Contributed by Mario Carneiro, 17-Jun-2014.)

Assertion

Ref	Expression
ismbf	|- ( F : A --> RR -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) )

Distinct variable groups:   x, F   x, A

Proof of Theorem ismbf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfdm 23395	. . 3 |- ( F e. MblFn -> dom F e. dom vol )
2		fdm 6051	. . . 4 |- ( F : A --> RR -> dom F = A )
3	2	eleq1d 2686	. . 3 |- ( F : A --> RR -> ( dom F e. dom vol <-> A e. dom vol ) )
4	1, 3	syl5ib 234	. 2 |- ( F : A --> RR -> ( F e. MblFn -> A e. dom vol ) )
5		ioomax 12248	. . . . 5 |- ( -oo (,) +oo ) = RR
6		ioorebas 12275	. . . . 5 |- ( -oo (,) +oo ) e. ran (,)
7	5, 6	eqeltrri 2698	. . . 4 |- RR e. ran (,)
8		imaeq2 5462	. . . . . 6 |- ( x = RR -> ( `' F " x ) = ( `' F " RR ) )
9	8	eleq1d 2686	. . . . 5 |- ( x = RR -> ( ( `' F " x ) e. dom vol <-> ( `' F " RR ) e. dom vol ) )
10	9	rspcv 3305	. . . 4 |- ( RR e. ran (,) -> ( A. x e. ran (,) ( `' F " x ) e. dom vol -> ( `' F " RR ) e. dom vol ) )
11	7, 10	ax-mp 5	. . 3 |- ( A. x e. ran (,) ( `' F " x ) e. dom vol -> ( `' F " RR ) e. dom vol )
12		fimacnv 6347	. . . 4 |- ( F : A --> RR -> ( `' F " RR ) = A )
13	12	eleq1d 2686	. . 3 |- ( F : A --> RR -> ( ( `' F " RR ) e. dom vol <-> A e. dom vol ) )
14	11, 13	syl5ib 234	. 2 |- ( F : A --> RR -> ( A. x e. ran (,) ( `' F " x ) e. dom vol -> A e. dom vol ) )
15		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( F : A --> RR /\ x e. A ) -> ( F ` x ) e. RR )
16	15	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( F : A --> RR /\ A e. dom vol ) /\ x e. A ) -> ( F ` x ) e. RR )
17	16	rered 13964	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A e. dom vol ) /\ x e. A ) -> ( Re ` ( F ` x ) ) = ( F ` x ) )
18	17	mpteq2dva 4744	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( x e. A |-> ( Re ` ( F ` x ) ) ) = ( x e. A |-> ( F ` x ) ) )
19	16	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A e. dom vol ) /\ x e. A ) -> ( F ` x ) e. CC )
20		simpl 473	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ A e. dom vol ) -> F : A --> RR )
21	20	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A e. dom vol ) -> F = ( x e. A |-> ( F ` x ) ) )
22		ref 13852	. . . . . . . . . . . . . 14 |- Re : CC --> RR
23	22	a1i 11	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ A e. dom vol ) -> Re : CC --> RR )
24	23	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A e. dom vol ) -> Re = ( y e. CC |-> ( Re ` y ) ) )
25		fveq2 6191	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( Re ` y ) = ( Re ` ( F ` x ) ) )
26	19, 21, 24, 25	fmptco 6396	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( Re o. F ) = ( x e. A |-> ( Re ` ( F ` x ) ) ) )
27	18, 26, 21	3eqtr4rd 2667	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A e. dom vol ) -> F = ( Re o. F ) )
28	27	cnveqd 5298	. . . . . . . . 9 |- ( ( F : A --> RR /\ A e. dom vol ) -> `' F = `' ( Re o. F ) )
29	28	imaeq1d 5465	. . . . . . . 8 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( `' F " x ) = ( `' ( Re o. F ) " x ) )
30	29	eleq1d 2686	. . . . . . 7 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( ( `' F " x ) e. dom vol <-> ( `' ( Re o. F ) " x ) e. dom vol ) )
31		imf 13853	. . . . . . . . . . . . . . . 16 |- Im : CC --> RR
32	31	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( F : A --> RR /\ A e. dom vol ) -> Im : CC --> RR )
33	32	feqmptd 6249	. . . . . . . . . . . . . 14 |- ( ( F : A --> RR /\ A e. dom vol ) -> Im = ( y e. CC |-> ( Im ` y ) ) )
34		fveq2 6191	. . . . . . . . . . . . . 14 |- ( y = ( F ` x ) -> ( Im ` y ) = ( Im ` ( F ` x ) ) )
35	19, 21, 33, 34	fmptco 6396	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( Im o. F ) = ( x e. A |-> ( Im ` ( F ` x ) ) ) )
36	16	reim0d 13965	. . . . . . . . . . . . . 14 |- ( ( ( F : A --> RR /\ A e. dom vol ) /\ x e. A ) -> ( Im ` ( F ` x ) ) = 0 )
37	36	mpteq2dva 4744	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( x e. A |-> ( Im ` ( F ` x ) ) ) = ( x e. A |-> 0 ) )
38	35, 37	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( Im o. F ) = ( x e. A |-> 0 ) )
39		fconstmpt 5163	. . . . . . . . . . . 12 |- ( A X. { 0 } ) = ( x e. A |-> 0 )
40	38, 39	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( Im o. F ) = ( A X. { 0 } ) )
41	40	cnveqd 5298	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A e. dom vol ) -> `' ( Im o. F ) = `' ( A X. { 0 } ) )
42	41	imaeq1d 5465	. . . . . . . . 9 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( `' ( Im o. F ) " x ) = ( `' ( A X. { 0 } ) " x ) )
43		simpr 477	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A e. dom vol ) -> A e. dom vol )
44		0re 10040	. . . . . . . . . 10 |- 0 e. RR
45		mbfconstlem 23396	. . . . . . . . . 10 |- ( ( A e. dom vol /\ 0 e. RR ) -> ( `' ( A X. { 0 } ) " x ) e. dom vol )
46	43, 44, 45	sylancl 694	. . . . . . . . 9 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( `' ( A X. { 0 } ) " x ) e. dom vol )
47	42, 46	eqeltrd 2701	. . . . . . . 8 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( `' ( Im o. F ) " x ) e. dom vol )
48	47	biantrud 528	. . . . . . 7 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( ( `' ( Re o. F ) " x ) e. dom vol <-> ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
49	30, 48	bitrd 268	. . . . . 6 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( ( `' F " x ) e. dom vol <-> ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
50	49	ralbidv 2986	. . . . 5 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( A. x e. ran (,) ( `' F " x ) e. dom vol <-> A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
51		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
52		fss 6056	. . . . . . . 8 |- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC )
53	51, 52	mpan2 707	. . . . . . 7 |- ( F : A --> RR -> F : A --> CC )
54		mblss 23299	. . . . . . 7 |- ( A e. dom vol -> A C_ RR )
55		cnex 10017	. . . . . . . 8 |- CC e. _V
56		reex 10027	. . . . . . . 8 |- RR e. _V
57		elpm2r 7875	. . . . . . . 8 |- ( ( ( CC e. _V /\ RR e. _V ) /\ ( F : A --> CC /\ A C_ RR ) ) -> F e. ( CC ^pm RR ) )
58	55, 56, 57	mpanl12 718	. . . . . . 7 |- ( ( F : A --> CC /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
59	53, 54, 58	syl2an 494	. . . . . 6 |- ( ( F : A --> RR /\ A e. dom vol ) -> F e. ( CC ^pm RR ) )
60	59	biantrurd 529	. . . . 5 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) <-> ( F e. ( CC ^pm RR ) /\ A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) ) )
61	50, 60	bitrd 268	. . . 4 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( A. x e. ran (,) ( `' F " x ) e. dom vol <-> ( F e. ( CC ^pm RR ) /\ A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) ) )
62		ismbf1 23393	. . . 4 |- ( F e. MblFn <-> ( F e. ( CC ^pm RR ) /\ A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
63	61, 62	syl6rbbr 279	. . 3 |- ( ( F : A --> RR /\ A e. dom vol ) -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) )
64	63	ex 450	. 2 |- ( F : A --> RR -> ( A e. dom vol -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) )
65	4, 14, 64	pm5.21ndd 369	1 |- ( F : A --> RR -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   -oocmnf 10072   (,)cioo 12175   Recre 13837   Imcim 13838   volcvol 23232  MblFncmbf 23383

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-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-fal 1489  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-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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  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-oi 8415  df-card 8765  df-cda 8990  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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388

This theorem is referenced by:  ismbfcn  23398  mbfima  23399  mbfid  23403  ismbfd  23407  mbfeqalem  23409  mbfres2  23412  mbfimaopnlem  23422  i1fd  23448  elmbfmvol2  30329  cnambfre  33458  mbf0  40173