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Theorem mbfimaicc 19517
Description: The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
mbfimaicc  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( B [,] C ) )  e. 
dom  vol )

Proof of Theorem mbfimaicc
StepHypRef Expression
1 iccssre 10984 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B [,] C
)  C_  RR )
21adantl 453 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  C_  RR )
3 dfss4 3567 . . . . . 6  |-  ( ( B [,] C ) 
C_  RR  <->  ( RR  \ 
( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
42, 3sylib 189 . . . . 5  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
5 difreicc 11020 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( RR  \  ( B [,] C ) )  =  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) )
65adantl 453 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( B [,] C
) )  =  ( (  -oo (,) B
)  u.  ( C (,)  +oo ) ) )
76difeq2d 3457 . . . . 5  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( RR  \  ( B [,] C ) ) )  =  ( RR 
\  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) )
84, 7eqtr3d 2469 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  =  ( RR  \  ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )
98imaeq2d 5195 . . 3  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( B [,] C ) )  =  ( `' F "
( RR  \  (
(  -oo (,) B )  u.  ( C (,)  +oo ) ) ) ) )
10 ffun 5585 . . . . . 6  |-  ( F : A --> RR  ->  Fun 
F )
11 funcnvcnv 5501 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
1210, 11syl 16 . . . . 5  |-  ( F : A --> RR  ->  Fun  `' `' F )
1312ad2antlr 708 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  Fun  `' `' F )
14 imadif 5520 . . . 4  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) ) )
1513, 14syl 16 . . 3  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( RR  \ 
( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) ) )
169, 15eqtrd 2467 . 2  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( B [,] C ) )  =  ( ( `' F " RR )  \  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) ) )
17 fimacnv 5854 . . . . . 6  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1817adantl 453 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  =  A )
19 mbfdm 19512 . . . . . 6  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
20 fdm 5587 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
2120eleq1d 2501 . . . . . . 7  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
2221biimpac 473 . . . . . 6  |-  ( ( dom  F  e.  dom  vol 
/\  F : A --> RR )  ->  A  e. 
dom  vol )
2319, 22sylan 458 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  A  e.  dom  vol )
2418, 23eqeltrd 2509 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  e.  dom  vol )
25 imaundi 5276 . . . . 5  |-  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  =  ( ( `' F " (  -oo (,) B
) )  u.  ( `' F " ( C (,)  +oo ) ) )
26 mbfima 19516 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " (  -oo (,) B ) )  e. 
dom  vol )
27 mbfima 19516 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( C (,)  +oo ) )  e. 
dom  vol )
28 unmbl 19424 . . . . . 6  |-  ( ( ( `' F "
(  -oo (,) B ) )  e.  dom  vol  /\  ( `' F "
( C (,)  +oo ) )  e.  dom  vol )  ->  ( ( `' F " (  -oo (,) B ) )  u.  ( `' F "
( C (,)  +oo ) ) )  e. 
dom  vol )
2926, 27, 28syl2anc 643 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F "
(  -oo (,) B ) )  u.  ( `' F " ( C (,)  +oo ) ) )  e.  dom  vol )
3025, 29syl5eqel 2519 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  e. 
dom  vol )
31 difmbl 19429 . . . 4  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) )  e.  dom  vol )  ->  ( ( `' F " RR )  \  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )  e.  dom  vol )
3224, 30, 31syl2anc 643 . . 3  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F " RR )  \  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )  e.  dom  vol )
3332adantr 452 . 2  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( ( `' F " RR ) 
\  ( `' F " ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) )  e.  dom  vol )
3416, 33eqeltrd 2509 1  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( `' F " ( B [,] C ) )  e. 
dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    C_ wss 3312   `'ccnv 4869   dom cdm 4870   "cima 4873   Fun wfun 5440   -->wf 5442  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   (,)cioo 10908   [,]cicc 10911   volcvol 19352  MblFncmbf 19498
This theorem is referenced by:  mbfimasn  19518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504
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