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Theorem mbfimaicc 18988
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 10731 . . . . . . 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 3403 . . . . . 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 10767 . . . . . . 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 3294 . . . . 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 2317 . . . 4  |-  ( ( ( F  e. MblFn  /\  F : A --> RR )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B [,] C )  =  ( RR  \  ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) ) )
98imaeq2d 5012 . . 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 5391 . . . . . 6  |-  ( F : A --> RR  ->  Fun 
F )
11 funcnvcnv 5308 . . . . . 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 5327 . . . 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 2315 . 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 5657 . . . . . 6  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1817adantl 452 . . . . 5  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  =  A )
19 mbfdm 18983 . . . . . 6  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
20 fdm 5393 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
2120eleq1d 2349 . . . . . . 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 2357 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " RR )  e.  dom  vol )
25 imaundi 5093 . . . . 5  |-  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  =  ( ( `' F " (  -oo (,) B
) )  u.  ( `' F " ( C (,)  +oo ) ) )
26 mbfima 18987 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " (  -oo (,) B ) )  e. 
dom  vol )
27 mbfima 18987 . . . . . 6  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( C (,)  +oo ) )  e. 
dom  vol )
28 unmbl 18895 . . . . . 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 2367 . . . 4  |-  ( ( F  e. MblFn  /\  F : A
--> RR )  ->  ( `' F " ( ( 
-oo (,) B )  u.  ( C (,)  +oo ) ) )  e. 
dom  vol )
31 difmbl 18900 . . . 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 2357 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 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    C_ wss 3152   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249   -->wf 5251  (class class class)co 5858   RRcr 8736    +oocpnf 8864    -oocmnf 8865   (,)cioo 10656   [,]cicc 10659   volcvol 18823  MblFncmbf 18969
This theorem is referenced by:  mbfimasn  18989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975
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