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Theorem mbfres2 22613
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 5108 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( B  u.  C )
)  =  ( F  |`  A ) )
3 mbfres2.1 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR )
4 ffn 5733 . . . . . . . . . . . 12  |-  ( F : A --> RR  ->  F  Fn  A )
5 fnresdm 5690 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
63, 4, 53syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  A )  =  F )
72, 6eqtr2d 2488 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( F  |`  ( B  u.  C
) ) )
87adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( F  |`  ( B  u.  C
) ) )
9 resundi 5121 . . . . . . . . 9  |-  ( F  |`  ( B  u.  C
) )  =  ( ( F  |`  B )  u.  ( F  |`  C ) )
108, 9syl6eq 2503 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( ( F  |`  B )  u.  ( F  |`  C ) ) )
1110cnveqd 5013 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  `' (
( F  |`  B )  u.  ( F  |`  C ) ) )
12 cnvun 5244 . . . . . . 7  |-  `' ( ( F  |`  B )  u.  ( F  |`  C ) )  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) )
1311, 12syl6eq 2503 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) )
1413imaeq1d 5170 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) "
x ) )
15 imaundir 5252 . . . . 5  |-  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )
1614, 15syl6eq 2503 . . . 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 3599 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
1918, 1syl5sseq 3482 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
203, 19fssresd 5755 . . . . . . . 8  |-  ( ph  ->  ( F  |`  B ) : B --> RR )
21 ismbf 22598 . . . . . . . 8  |-  ( ( F  |`  B ) : B --> RR  ->  (
( F  |`  B )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol ) )
2220, 21syl 17 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  B )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  B )
" x )  e. 
dom  vol ) )
2317, 22mpbid 214 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol )
2423r19.21bi 2759 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  B ) " x
)  e.  dom  vol )
25 mbfres2.3 . . . . . . 7  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
26 ssun2 3600 . . . . . . . . . 10  |-  C  C_  ( B  u.  C
)
2726, 1syl5sseq 3482 . . . . . . . . 9  |-  ( ph  ->  C  C_  A )
283, 27fssresd 5755 . . . . . . . 8  |-  ( ph  ->  ( F  |`  C ) : C --> RR )
29 ismbf 22598 . . . . . . . 8  |-  ( ( F  |`  C ) : C --> RR  ->  (
( F  |`  C )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol ) )
3028, 29syl 17 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  C )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  C )
" x )  e. 
dom  vol ) )
3125, 30mpbid 214 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol )
3231r19.21bi 2759 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  C ) " x
)  e.  dom  vol )
33 unmbl 22503 . . . . 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 667 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  (
( `' ( F  |`  B ) " x
)  u.  ( `' ( F  |`  C )
" x ) )  e.  dom  vol )
3516, 34eqeltrd 2531 . . 3  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
3635ralrimiva 2804 . 2  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
37 ismbf 22598 . . 3  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
383, 37syl 17 . 2  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
3936, 38mpbird 236 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739    u. cun 3404   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    Fn wfn 5580   -->wf 5581   RRcr 9543   (,)cioo 11642   volcvol 22427  MblFncmbf 22584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xadd 11417  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-xmet 18975  df-met 18976  df-ovol 22428  df-vol 22430  df-mbf 22589
This theorem is referenced by:  mbfss  22614  mbfresfi  31999  mbfposadd  32000  mbfres2cn  37845
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