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Theorem mbfres2 21239
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 5208 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( B  u.  C )
)  =  ( F  |`  A ) )
3 mbfres2.1 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR )
4 ffn 5657 . . . . . . . . . . . 12  |-  ( F : A --> RR  ->  F  Fn  A )
5 fnresdm 5618 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
63, 4, 53syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  A )  =  F )
72, 6eqtr2d 2493 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( F  |`  ( B  u.  C
) ) )
87adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( F  |`  ( B  u.  C
) ) )
9 resundi 5222 . . . . . . . . 9  |-  ( F  |`  ( B  u.  C
) )  =  ( ( F  |`  B )  u.  ( F  |`  C ) )
108, 9syl6eq 2508 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( ( F  |`  B )  u.  ( F  |`  C ) ) )
1110cnveqd 5113 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  `' (
( F  |`  B )  u.  ( F  |`  C ) ) )
12 cnvun 5340 . . . . . . 7  |-  `' ( ( F  |`  B )  u.  ( F  |`  C ) )  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) )
1311, 12syl6eq 2508 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) )
1413imaeq1d 5266 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) "
x ) )
15 imaundir 5348 . . . . 5  |-  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )
1614, 15syl6eq 2508 . . . 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 3617 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
1918, 1syl5sseq 3502 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
20 fssres 5676 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  B  C_  A )  -> 
( F  |`  B ) : B --> RR )
213, 19, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F  |`  B ) : B --> RR )
22 ismbf 21224 . . . . . . . 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 210 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol )
2524r19.21bi 2910 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  B ) " x
)  e.  dom  vol )
26 mbfres2.3 . . . . . . 7  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
27 ssun2 3618 . . . . . . . . . 10  |-  C  C_  ( B  u.  C
)
2827, 1syl5sseq 3502 . . . . . . . . 9  |-  ( ph  ->  C  C_  A )
29 fssres 5676 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  C  C_  A )  -> 
( F  |`  C ) : C --> RR )
303, 28, 29syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F  |`  C ) : C --> RR )
31 ismbf 21224 . . . . . . . 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 210 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol )
3433r19.21bi 2910 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  C ) " x
)  e.  dom  vol )
35 unmbl 21135 . . . . 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 661 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  (
( `' ( F  |`  B ) " x
)  u.  ( `' ( F  |`  C )
" x ) )  e.  dom  vol )
3716, 36eqeltrd 2539 . . 3  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
3837ralrimiva 2822 . 2  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
39 ismbf 21224 . . 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 232 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    u. cun 3424    C_ wss 3426   `'ccnv 4937   dom cdm 4938   ran crn 4939    |` cres 4940   "cima 4941    Fn wfn 5511   -->wf 5512   RRcr 9382   (,)cioo 11401   volcvol 21063  MblFncmbf 21210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-xadd 11191  df-ioo 11405  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-xmet 17919  df-met 17920  df-ovol 21064  df-vol 21065  df-mbf 21215
This theorem is referenced by:  mbfss  21240  mbfresfi  28576  mbfposadd  28577
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