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Theorem i1fima2 19573
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 19572 . . . 4  |-  ( F  e.  dom  S.1  ->  ( `' F " A )  e.  dom  vol )
21adantr 453 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  e. 
dom  vol )
3 mblvol 19428 . . 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 19570 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
65adantr 453 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  F : RR
--> RR )
7 ffun 5595 . . . . . 6  |-  ( F : RR --> RR  ->  Fun 
F )
8 inpreima 5859 . . . . . 6  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
96, 7, 83syl 19 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
10 cnvimass 5226 . . . . . . 7  |-  ( `' F " A ) 
C_  dom  F
11 cnvimarndm 5227 . . . . . . 7  |-  ( `' F " ran  F
)  =  dom  F
1210, 11sseqtr4i 3383 . . . . . 6  |-  ( `' F " A ) 
C_  ( `' F " ran  F )
13 df-ss 3336 . . . . . 6  |-  ( ( `' F " A ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A ) )
1412, 13mpbi 201 . . . . 5  |-  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A )
159, 14syl6req 2487 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
16 inss1 3563 . . . . . . . . . 10  |-  ( A  i^i  ran  F )  C_  A
1716sseli 3346 . . . . . . . . 9  |-  ( 0  e.  ( A  i^i  ran 
F )  ->  0  e.  A )
1817con3i 130 . . . . . . . 8  |-  ( -.  0  e.  A  ->  -.  0  e.  ( A  i^i  ran  F )
)
1918adantl 454 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  -.  0  e.  ( A  i^i  ran  F ) )
20 disjsn 3870 . . . . . . 7  |-  ( ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( A  i^i  ran  F ) )
2119, 20sylibr 205 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/) )
22 inss2 3564 . . . . . . . . 9  |-  ( A  i^i  ran  F )  C_ 
ran  F
23 frn 5599 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
245, 23syl 16 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ran 
F  C_  RR )
2522, 24syl5ss 3361 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( A  i^i  ran  F
)  C_  RR )
2625adantr 453 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  RR )
27 reldisj 3673 . . . . . . 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 203 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) )
30 imass2 5242 . . . . 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 3384 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
33 i1fima 19572 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol )
3433adantr 453 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  e.  dom  vol )
35 mblss 19429 . . . 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 19428 . . . . 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 19568 . . . . . . 7  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
4039simprbi 452 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F : RR --> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
4140simp3d 972 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
4241adantr 453 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR )
4338, 42eqeltrrd 2513 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol * `
 ( `' F " ( RR  \  {
0 } ) ) )  e.  RR )
44 ovolsscl 19384 . . 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 1185 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol * `
 ( `' F " A ) )  e.  RR )
464, 45eqeltrd 2512 1  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883   Fun wfun 5450   -->wf 5452   ` cfv 5456   Fincfn 7111   RRcr 8991   0cc0 8992   vol *covol 19361   volcvol 19362  MblFncmbf 19508   S.1citg1 19509
This theorem is referenced by:  i1fima2sn  19574  i1f0rn  19576  itg2addnclem  26258  itg2addnclem2  26259  ftc1anclem3  26284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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