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Theorem mbfimaicc 19004
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 10747 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B [,] C
)  C_  RR )
21adantl 452 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  C_  RR )
3 dfss4 3416 . . . . . 6  |-  ( ( B [,] C ) 
C_  RR  <->  ( RR  \ 
( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
42, 3sylib 188 . . . . 5  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( RR  \  ( B [,] C ) ) )  =  ( B [,] C ) )
5 difreicc 10783 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( RR  \  ( B [,] C ) )  =  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) )
65adantl 452 . . . . . 6  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( RR  \  ( B [,] C
) )  =  ( (  -oo (,) B
)  u.  ( C (,)  +oo ) ) )
76difeq2d 3307 . . . . 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 2330 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  =  ( RR  \  ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )
98imaeq2d 5028 . . 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 5407 . . . . . 6  |-  ( F : A --> RR  ->  Fun 
F )
11 funcnvcnv 5324 . . . . . 6  |-  ( Fun 
F  ->  Fun  `' `' F )
1210, 11syl 15 . . . . 5  |-  ( F : A --> RR  ->  Fun  `' `' F )
1312ad2antlr 707 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  Fun  `' `' F )
14 imadif 5343 . . . 4  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( (  -oo (,) B )  u.  ( C (,)  +oo ) ) ) ) )
1513, 14syl 15 . . 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 2328 . 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 5673 . . . . . 6  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1817adantl 452 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  =  A )
19 mbfdm 18999 . . . . . 6  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
20 fdm 5409 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
2120eleq1d 2362 . . . . . . 7  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
2221biimpac 472 . . . . . 6  |-  ( ( dom  F  e.  dom  vol 
/\  F : A --> RR )  ->  A  e. 
dom  vol )
2319, 22sylan 457 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  A  e.  dom  vol )
2418, 23eqeltrd 2370 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  e.  dom  vol )
25 imaundi 5109 . . . . 5  |-  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  =  ( ( `' F " (  -oo (,) B
) )  u.  ( `' F " ( C (,)  +oo ) ) )
26 mbfima 19003 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " (  -oo (,) B ) )  e. 
dom  vol )
27 mbfima 19003 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( C (,)  +oo ) )  e. 
dom  vol )
28 unmbl 18911 . . . . . 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 642 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F "
(  -oo (,) B ) )  u.  ( `' F " ( C (,)  +oo ) ) )  e.  dom  vol )
3025, 29syl5eqel 2380 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  e. 
dom  vol )
31 difmbl 18916 . . . 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 642 . . 3  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  (
( `' F " RR )  \  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )  e.  dom  vol )
3332adantr 451 . 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 2370 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 358    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   -->wf 5267  (class class class)co 5874   RRcr 8752    +oocpnf 8880    -oocmnf 8881   (,)cioo 10672   [,]cicc 10675   volcvol 18839  MblFncmbf 18985
This theorem is referenced by:  mbfimasn  19005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991
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