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Theorem i1fima2 21179
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 21178 . . . 4  |-  ( F  e.  dom  S.1  ->  ( `' F " A )  e.  dom  vol )
21adantr 465 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  e. 
dom  vol )
3 mblvol 21035 . . 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 21176 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
65adantr 465 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  F : RR
--> RR )
7 ffun 5582 . . . . . 6  |-  ( F : RR --> RR  ->  Fun 
F )
8 inpreima 5851 . . . . . 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 5210 . . . . . . 7  |-  ( `' F " A ) 
C_  dom  F
11 cnvimarndm 5211 . . . . . . 7  |-  ( `' F " ran  F
)  =  dom  F
1210, 11sseqtr4i 3410 . . . . . 6  |-  ( `' F " A ) 
C_  ( `' F " ran  F )
13 df-ss 3363 . . . . . 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 2492 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
16 inss1 3591 . . . . . . . . . 10  |-  ( A  i^i  ran  F )  C_  A
1716sseli 3373 . . . . . . . . 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 466 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  -.  0  e.  ( A  i^i  ran  F ) )
20 disjsn 3957 . . . . . . 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 3592 . . . . . . . . 9  |-  ( A  i^i  ran  F )  C_ 
ran  F
23 frn 5586 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
245, 23syl 16 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ran 
F  C_  RR )
2522, 24syl5ss 3388 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( A  i^i  ran  F
)  C_  RR )
2625adantr 465 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  RR )
27 reldisj 3743 . . . . . . 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 5225 . . . . 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 3411 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
33 i1fima 21178 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol )
3433adantr 465 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  e.  dom  vol )
35 mblss 21036 . . . 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 21035 . . . . 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 21174 . . . . . . 7  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
4039simprbi 464 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F : RR --> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
4140simp3d 1002 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
4241adantr 465 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR )
4338, 42eqeltrrd 2518 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " ( RR  \  {
0 } ) ) )  e.  RR )
44 ovolsscl 20991 . . 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 1218 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " A ) )  e.  RR )
464, 45eqeltrd 2517 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864   Fun wfun 5433   -->wf 5435   ` cfv 5439   Fincfn 7331   RRcr 9302   0cc0 9303   vol*covol 20968   volcvol 20969  MblFncmbf 21116   S.1citg1 21117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-xadd 11111  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-xmet 17832  df-met 17833  df-ovol 20970  df-vol 20971  df-mbf 21121  df-itg1 21122
This theorem is referenced by:  i1fima2sn  21180  i1f0rn  21182  itg2addnclem  28469  itg2addnclem2  28470  ftc1anclem3  28495
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