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Theorem mbfres2 19000
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 4955 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( B  u.  C )
)  =  ( F  |`  A ) )
3 mbfres2.1 . . . . . . . . . . . 12  |-  ( ph  ->  F : A --> RR )
4 ffn 5389 . . . . . . . . . . . 12  |-  ( F : A --> RR  ->  F  Fn  A )
5 fnresdm 5353 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
63, 4, 53syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  A )  =  F )
72, 6eqtr2d 2316 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( F  |`  ( B  u.  C
) ) )
87adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( F  |`  ( B  u.  C
) ) )
9 resundi 4969 . . . . . . . . 9  |-  ( F  |`  ( B  u.  C
) )  =  ( ( F  |`  B )  u.  ( F  |`  C ) )
108, 9syl6eq 2331 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  F  =  ( ( F  |`  B )  u.  ( F  |`  C ) ) )
1110cnveqd 4857 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  `' (
( F  |`  B )  u.  ( F  |`  C ) ) )
12 cnvun 5086 . . . . . . 7  |-  `' ( ( F  |`  B )  u.  ( F  |`  C ) )  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) )
1311, 12syl6eq 2331 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  `' F  =  ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) )
1413imaeq1d 5011 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  =  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) "
x ) )
15 imaundir 5094 . . . . 5  |-  ( ( `' ( F  |`  B )  u.  `' ( F  |`  C ) ) " x )  =  ( ( `' ( F  |`  B )
" x )  u.  ( `' ( F  |`  C ) " x
) )
1614, 15syl6eq 2331 . . . 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 3338 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
1918, 1syl5sseq 3226 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
20 fssres 5408 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  B  C_  A )  -> 
( F  |`  B ) : B --> RR )
213, 19, 20syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( F  |`  B ) : B --> RR )
22 ismbf 18985 . . . . . . . 8  |-  ( ( F  |`  B ) : B --> RR  ->  (
( F  |`  B )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol ) )
2321, 22syl 15 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  B )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  B )
" x )  e. 
dom  vol ) )
2417, 23mpbid 201 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  B ) " x
)  e.  dom  vol )
2524r19.21bi 2641 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  B ) " x
)  e.  dom  vol )
26 mbfres2.3 . . . . . . 7  |-  ( ph  ->  ( F  |`  C )  e. MblFn )
27 ssun2 3339 . . . . . . . . . 10  |-  C  C_  ( B  u.  C
)
2827, 1syl5sseq 3226 . . . . . . . . 9  |-  ( ph  ->  C  C_  A )
29 fssres 5408 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  C  C_  A )  -> 
( F  |`  C ) : C --> RR )
303, 28, 29syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( F  |`  C ) : C --> RR )
31 ismbf 18985 . . . . . . . 8  |-  ( ( F  |`  C ) : C --> RR  ->  (
( F  |`  C )  e. MblFn 
<-> 
A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol ) )
3230, 31syl 15 . . . . . . 7  |-  ( ph  ->  ( ( F  |`  C )  e. MblFn  <->  A. x  e.  ran  (,) ( `' ( F  |`  C )
" x )  e. 
dom  vol ) )
3326, 32mpbid 201 . . . . . 6  |-  ( ph  ->  A. x  e.  ran  (,) ( `' ( F  |`  C ) " x
)  e.  dom  vol )
3433r19.21bi 2641 . . . . 5  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' ( F  |`  C ) " x
)  e.  dom  vol )
35 unmbl 18895 . . . . 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 642 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  (
( `' ( F  |`  B ) " x
)  u.  ( `' ( F  |`  C )
" x ) )  e.  dom  vol )
3716, 36eqeltrd 2357 . . 3  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
3837ralrimiva 2626 . 2  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
39 ismbf 18985 . . 3  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
403, 39syl 15 . 2  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
4138, 40mpbird 223 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   RRcr 8736   (,)cioo 10656   volcvol 18823  MblFncmbf 18969
This theorem is referenced by:  mbfss  19001
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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