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Theorem mbfres2 19404
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 5086 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( B  u.  C )
)  =  ( F  |`  A ) )
3 mbfres2.1 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR )
4 ffn 5531 . . . . . . . . . . . 12  |-  ( F : A --> RR  ->  F  Fn  A )
5 fnresdm 5494 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
63, 4, 53syl 19 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  A )  =  F )
72, 6eqtr2d 2420 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( F  |`  ( B  u.  C
) ) )
87adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( F  |`  ( B  u.  C
) ) )
9 resundi 5100 . . . . . . . . 9  |-  ( F  |`  ( B  u.  C
) )  =  ( ( F  |`  B )  u.  ( F  |`  C ) )
108, 9syl6eq 2435 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( ( F  |`  B )  u.  ( F  |`  C ) ) )
1110cnveqd 4988 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  `' (
( F  |`  B )  u.  ( F  |`  C ) ) )
12 cnvun 5217 . . . . . . 7  |-  `' ( ( F  |`  B )  u.  ( F  |`  C ) )  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) )
1311, 12syl6eq 2435 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) )
1413imaeq1d 5142 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) "
x ) )
15 imaundir 5225 . . . . 5  |-  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )
1614, 15syl6eq 2435 . . . 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 3453 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
1918, 1syl5sseq 3339 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
20 fssres 5550 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  B  C_  A )  -> 
( F  |`  B ) : B --> RR )
213, 19, 20syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( F  |`  B ) : B --> RR )
22 ismbf 19389 . . . . . . . 8  |-  ( ( F  |`  B ) : B --> RR  ->  (
( F  |`  B )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol ) )
2321, 22syl 16 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  B )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  B )
" x )  e. 
dom  vol ) )
2417, 23mpbid 202 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol )
2524r19.21bi 2747 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  B ) " x
)  e.  dom  vol )
26 mbfres2.3 . . . . . . 7  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
27 ssun2 3454 . . . . . . . . . 10  |-  C  C_  ( B  u.  C
)
2827, 1syl5sseq 3339 . . . . . . . . 9  |-  ( ph  ->  C  C_  A )
29 fssres 5550 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  C  C_  A )  -> 
( F  |`  C ) : C --> RR )
303, 28, 29syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( F  |`  C ) : C --> RR )
31 ismbf 19389 . . . . . . . 8  |-  ( ( F  |`  C ) : C --> RR  ->  (
( F  |`  C )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol ) )
3230, 31syl 16 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  C )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  C )
" x )  e. 
dom  vol ) )
3326, 32mpbid 202 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol )
3433r19.21bi 2747 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  C ) " x
)  e.  dom  vol )
35 unmbl 19299 . . . . 5  |-  ( ( ( `' ( F  |`  B ) " x
)  e.  dom  vol  /\  ( `' ( F  |`  C ) " x
)  e.  dom  vol )  ->  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )  e.  dom  vol )
3625, 34, 35syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  (
( `' ( F  |`  B ) " x
)  u.  ( `' ( F  |`  C )
" x ) )  e.  dom  vol )
3716, 36eqeltrd 2461 . . 3  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
3837ralrimiva 2732 . 2  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
39 ismbf 19389 . . 3  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
403, 39syl 16 . 2  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
4138, 40mpbird 224 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    u. cun 3261    C_ wss 3263   `'ccnv 4817   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821    Fn wfn 5389   -->wf 5390   RRcr 8922   (,)cioo 10848   volcvol 19227  MblFncmbf 19373
This theorem is referenced by:  mbfss  19405  itgaddnclem2  25964
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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