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Theorem mbfimaicc 19392
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 10924 . . . . . . 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 3518 . . . . . 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 10960 . . . . . . 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 3408 . . . . 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 2421 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  =  ( RR  \  ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )
98imaeq2d 5143 . . 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 5533 . . . . . 6  |-  ( F : A --> RR  ->  Fun 
F )
11 funcnvcnv 5449 . . . . . 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 5468 . . . 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 2419 . 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 5801 . . . . . 6  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1817adantl 453 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  =  A )
19 mbfdm 19387 . . . . . 6  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
20 fdm 5535 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
2120eleq1d 2453 . . . . . . 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 2461 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  e.  dom  vol )
25 imaundi 5224 . . . . 5  |-  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  =  ( ( `' F " (  -oo (,) B
) )  u.  ( `' F " ( C (,)  +oo ) ) )
26 mbfima 19391 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " (  -oo (,) B ) )  e. 
dom  vol )
27 mbfima 19391 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( C (,)  +oo ) )  e. 
dom  vol )
28 unmbl 19299 . . . . . 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 2471 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  e. 
dom  vol )
31 difmbl 19304 . . . 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 2461 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 1649    e. wcel 1717    \ cdif 3260    u. cun 3261    C_ wss 3263   `'ccnv 4817   dom cdm 4818   "cima 4821   Fun wfun 5388   -->wf 5390  (class class class)co 6020   RRcr 8922    +oocpnf 9050    -oocmnf 9051   (,)cioo 10848   [,]cicc 10851   volcvol 19227  MblFncmbf 19373
This theorem is referenced by:  mbfimasn  19393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-xadd 10643  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-xmet 16619  df-met 16620  df-ovol 19228  df-vol 19229  df-mbf 19379
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