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Theorem mbfimaicc 19528
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 10997 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B [,] C
)  C_  RR )
21adantl 454 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  C_  RR )
3 dfss4 3577 . . . . . 6  |-  ( ( B [,] C ) 
C_  RR  <->  ( RR  \ 
( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
42, 3sylib 190 . . . . 5  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
5 difreicc 11033 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( RR  \  ( B [,] C ) )  =  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) )
65adantl 454 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( B [,] C
) )  =  ( (  -oo (,) B
)  u.  ( C (,)  +oo ) ) )
76difeq2d 3467 . . . . 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 2472 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  =  ( RR  \  ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )
98imaeq2d 5206 . . 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 5596 . . . . . 6  |-  ( F : A --> RR  ->  Fun 
F )
11 funcnvcnv 5512 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
1210, 11syl 16 . . . . 5  |-  ( F : A --> RR  ->  Fun  `' `' F )
1312ad2antlr 709 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  Fun  `' `' F )
14 imadif 5531 . . . 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 2470 . 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 5865 . . . . . 6  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1817adantl 454 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  =  A )
19 mbfdm 19523 . . . . . 6  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
20 fdm 5598 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
2120eleq1d 2504 . . . . . . 7  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
2221biimpac 474 . . . . . 6  |-  ( ( dom  F  e.  dom  vol 
/\  F : A --> RR )  ->  A  e. 
dom  vol )
2319, 22sylan 459 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  A  e.  dom  vol )
2418, 23eqeltrd 2512 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  e.  dom  vol )
25 imaundi 5287 . . . . 5  |-  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  =  ( ( `' F " (  -oo (,) B
) )  u.  ( `' F " ( C (,)  +oo ) ) )
26 mbfima 19527 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " (  -oo (,) B ) )  e. 
dom  vol )
27 mbfima 19527 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( C (,)  +oo ) )  e. 
dom  vol )
28 unmbl 19437 . . . . . 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 644 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F "
(  -oo (,) B ) )  u.  ( `' F " ( C (,)  +oo ) ) )  e.  dom  vol )
3025, 29syl5eqel 2522 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  e. 
dom  vol )
31 difmbl 19442 . . . 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 644 . . 3  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F " RR )  \  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )  e.  dom  vol )
3332adantr 453 . 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 2512 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 360    = wceq 1653    e. wcel 1726    \ cdif 3319    u. cun 3320    C_ wss 3322   `'ccnv 4880   dom cdm 4881   "cima 4884   Fun wfun 5451   -->wf 5453  (class class class)co 6084   RRcr 8994    +oocpnf 9122    -oocmnf 9123   (,)cioo 10921   [,]cicc 10924   volcvol 19365  MblFncmbf 19511
This theorem is referenced by:  mbfimasn  19529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xadd 10716  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-xmet 16700  df-met 16701  df-ovol 19366  df-vol 19367  df-mbf 19516
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