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Theorem i1fima2 22171
Description: Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
i1fima2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  e.  RR )

Proof of Theorem i1fima2
StepHypRef Expression
1 i1fima 22170 . . . 4  |-  ( F  e.  dom  S.1  ->  ( `' F " A )  e.  dom  vol )
21adantr 463 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  e. 
dom  vol )
3 mblvol 22026 . . 3  |-  ( ( `' F " A )  e.  dom  vol  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
42, 3syl 16 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
5 i1ff 22168 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
65adantr 463 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  F : RR
--> RR )
7 ffun 5641 . . . . . 6  |-  ( F : RR --> RR  ->  Fun 
F )
8 inpreima 5916 . . . . . 6  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
96, 7, 83syl 20 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
10 cnvimass 5269 . . . . . . 7  |-  ( `' F " A ) 
C_  dom  F
11 cnvimarndm 5270 . . . . . . 7  |-  ( `' F " ran  F
)  =  dom  F
1210, 11sseqtr4i 3450 . . . . . 6  |-  ( `' F " A ) 
C_  ( `' F " ran  F )
13 df-ss 3403 . . . . . 6  |-  ( ( `' F " A ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A ) )
1412, 13mpbi 208 . . . . 5  |-  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A )
159, 14syl6req 2440 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
16 inss1 3632 . . . . . . . . . 10  |-  ( A  i^i  ran  F )  C_  A
1716sseli 3413 . . . . . . . . 9  |-  ( 0  e.  ( A  i^i  ran 
F )  ->  0  e.  A )
1817con3i 135 . . . . . . . 8  |-  ( -.  0  e.  A  ->  -.  0  e.  ( A  i^i  ran  F )
)
1918adantl 464 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  -.  0  e.  ( A  i^i  ran  F ) )
20 disjsn 4004 . . . . . . 7  |-  ( ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( A  i^i  ran  F ) )
2119, 20sylibr 212 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/) )
22 inss2 3633 . . . . . . . . 9  |-  ( A  i^i  ran  F )  C_ 
ran  F
23 frn 5645 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
245, 23syl 16 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ran 
F  C_  RR )
2522, 24syl5ss 3428 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( A  i^i  ran  F
)  C_  RR )
2625adantr 463 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  RR )
27 reldisj 3786 . . . . . . 7  |-  ( ( A  i^i  ran  F
)  C_  RR  ->  ( ( ( A  i^i  ran 
F )  i^i  {
0 } )  =  (/) 
<->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) ) )
2826, 27syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( (
( A  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  ( A  i^i  ran  F )  C_  ( RR  \  {
0 } ) ) )
2921, 28mpbid 210 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) )
30 imass2 5284 . . . . 5  |-  ( ( A  i^i  ran  F
)  C_  ( RR  \  { 0 } )  ->  ( `' F " ( A  i^i  ran  F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3129, 30syl 16 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3215, 31eqsstrd 3451 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
33 i1fima 22170 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol )
3433adantr 463 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  e.  dom  vol )
35 mblss 22027 . . . 4  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( `' F "
( RR  \  {
0 } ) ) 
C_  RR )
3634, 35syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  C_  RR )
37 mblvol 22026 . . . . 5  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  =  ( vol* `  ( `' F " ( RR 
\  { 0 } ) ) ) )
3834, 37syl 16 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  =  ( vol* `  ( `' F " ( RR  \  { 0 } ) ) ) )
39 isi1f 22166 . . . . . . 7  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
4039simprbi 462 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F : RR --> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
4140simp3d 1008 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
4241adantr 463 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR )
4338, 42eqeltrrd 2471 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " ( RR  \  {
0 } ) ) )  e.  RR )
44 ovolsscl 21982 . . 3  |-  ( ( ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) )  /\  ( `' F " ( RR 
\  { 0 } ) )  C_  RR  /\  ( vol* `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )  ->  ( vol* `  ( `' F " A ) )  e.  RR )
4532, 36, 43, 44syl3anc 1226 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " A ) )  e.  RR )
464, 45eqeltrd 2470 1  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    \ cdif 3386    i^i cin 3388    C_ wss 3389   (/)c0 3711   {csn 3944   `'ccnv 4912   dom cdm 4913   ran crn 4914   "cima 4916   Fun wfun 5490   -->wf 5492   ` cfv 5496   Fincfn 7435   RRcr 9402   0cc0 9403   vol*covol 21959   volcvol 21960  MblFncmbf 22108   S.1citg1 22109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
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 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xadd 11240  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511  df-xmet 18525  df-met 18526  df-ovol 21961  df-vol 21962  df-mbf 22113  df-itg1 22114
This theorem is referenced by:  i1fima2sn  22172  i1f0rn  22174  itg2addnclem  30232  itg2addnclem2  30233  ftc1anclem3  30258
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