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Theorem mbfres2 22160
Description: Measurability of a piecewise function: if  F is measurable on subsets  B and  C of its domain, and these pieces make up all of  A, then  F is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
mbfres2.1  |-  ( ph  ->  F : A --> RR )
mbfres2.2  |-  ( ph  ->  ( F  |`  B )  e. MblFn )
mbfres2.3  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
mbfres2.4  |-  ( ph  ->  ( B  u.  C
)  =  A )
Assertion
Ref Expression
mbfres2  |-  ( ph  ->  F  e. MblFn )

Proof of Theorem mbfres2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mbfres2.4 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  u.  C
)  =  A )
21reseq2d 5203 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( B  u.  C )
)  =  ( F  |`  A ) )
3 mbfres2.1 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR )
4 ffn 5656 . . . . . . . . . . . 12  |-  ( F : A --> RR  ->  F  Fn  A )
5 fnresdm 5615 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
63, 4, 53syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  A )  =  F )
72, 6eqtr2d 2438 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( F  |`  ( B  u.  C
) ) )
87adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( F  |`  ( B  u.  C
) ) )
9 resundi 5216 . . . . . . . . 9  |-  ( F  |`  ( B  u.  C
) )  =  ( ( F  |`  B )  u.  ( F  |`  C ) )
108, 9syl6eq 2453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( ( F  |`  B )  u.  ( F  |`  C ) ) )
1110cnveqd 5108 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  `' (
( F  |`  B )  u.  ( F  |`  C ) ) )
12 cnvun 5338 . . . . . . 7  |-  `' ( ( F  |`  B )  u.  ( F  |`  C ) )  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) )
1311, 12syl6eq 2453 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) )
1413imaeq1d 5265 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) "
x ) )
15 imaundir 5346 . . . . 5  |-  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )
1614, 15syl6eq 2453 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) ) )
17 mbfres2.2 . . . . . . 7  |-  ( ph  ->  ( F  |`  B )  e. MblFn )
18 ssun1 3598 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
1918, 1syl5sseq 3482 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
203, 19fssresd 5677 . . . . . . . 8  |-  ( ph  ->  ( F  |`  B ) : B --> RR )
21 ismbf 22145 . . . . . . . 8  |-  ( ( F  |`  B ) : B --> RR  ->  (
( F  |`  B )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol ) )
2220, 21syl 16 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  B )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  B )
" x )  e. 
dom  vol ) )
2317, 22mpbid 210 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol )
2423r19.21bi 2765 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  B ) " x
)  e.  dom  vol )
25 mbfres2.3 . . . . . . 7  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
26 ssun2 3599 . . . . . . . . . 10  |-  C  C_  ( B  u.  C
)
2726, 1syl5sseq 3482 . . . . . . . . 9  |-  ( ph  ->  C  C_  A )
283, 27fssresd 5677 . . . . . . . 8  |-  ( ph  ->  ( F  |`  C ) : C --> RR )
29 ismbf 22145 . . . . . . . 8  |-  ( ( F  |`  C ) : C --> RR  ->  (
( F  |`  C )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol ) )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  C )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  C )
" x )  e. 
dom  vol ) )
3125, 30mpbid 210 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol )
3231r19.21bi 2765 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  C ) " x
)  e.  dom  vol )
33 unmbl 22056 . . . . 5  |-  ( ( ( `' ( F  |`  B ) " x
)  e.  dom  vol  /\  ( `' ( F  |`  C ) " x
)  e.  dom  vol )  ->  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )  e.  dom  vol )
3424, 32, 33syl2anc 659 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  (
( `' ( F  |`  B ) " x
)  u.  ( `' ( F  |`  C )
" x ) )  e.  dom  vol )
3516, 34eqeltrd 2484 . . 3  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
3635ralrimiva 2810 . 2  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
37 ismbf 22145 . . 3  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
383, 37syl 16 . 2  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
3936, 38mpbird 232 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746    u. cun 3404   `'ccnv 4929   dom cdm 4930   ran crn 4931    |` cres 4932   "cima 4933    Fn wfn 5508   -->wf 5509   RRcr 9424   (,)cioo 11472   volcvol 21983  MblFncmbf 22131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-inf2 7994  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-2o 7071  df-oadd 7074  df-er 7251  df-map 7362  df-pm 7363  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-sup 7838  df-oi 7872  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-n0 10735  df-z 10804  df-uz 11024  df-q 11124  df-rp 11162  df-xadd 11262  df-ioo 11476  df-ico 11478  df-icc 11479  df-fz 11616  df-fzo 11740  df-fl 11851  df-seq 12034  df-exp 12093  df-hash 12331  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-clim 13336  df-sum 13534  df-xmet 18548  df-met 18549  df-ovol 21984  df-vol 21985  df-mbf 22136
This theorem is referenced by:  mbfss  22161  mbfresfi  30266  mbfposadd  30267  mbfres2cn  31962
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