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Theorem i1fima2 22630
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 22629 . . . 4  |-  ( F  e.  dom  S.1  ->  ( `' F " A )  e.  dom  vol )
21adantr 467 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  e. 
dom  vol )
3 mblvol 22477 . . 3  |-  ( ( `' F " A )  e.  dom  vol  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
42, 3syl 17 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
5 i1ff 22627 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
65adantr 467 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  F : RR
--> RR )
7 ffun 5729 . . . . . 6  |-  ( F : RR --> RR  ->  Fun 
F )
8 inpreima 6005 . . . . . 6  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
96, 7, 83syl 18 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
10 cnvimass 5187 . . . . . . 7  |-  ( `' F " A ) 
C_  dom  F
11 cnvimarndm 5188 . . . . . . 7  |-  ( `' F " ran  F
)  =  dom  F
1210, 11sseqtr4i 3464 . . . . . 6  |-  ( `' F " A ) 
C_  ( `' F " ran  F )
13 df-ss 3417 . . . . . 6  |-  ( ( `' F " A ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A ) )
1412, 13mpbi 212 . . . . 5  |-  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A )
159, 14syl6req 2501 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
16 inss1 3651 . . . . . . . . . 10  |-  ( A  i^i  ran  F )  C_  A
1716sseli 3427 . . . . . . . . 9  |-  ( 0  e.  ( A  i^i  ran 
F )  ->  0  e.  A )
1817con3i 141 . . . . . . . 8  |-  ( -.  0  e.  A  ->  -.  0  e.  ( A  i^i  ran  F )
)
1918adantl 468 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  -.  0  e.  ( A  i^i  ran  F ) )
20 disjsn 4031 . . . . . . 7  |-  ( ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( A  i^i  ran  F ) )
2119, 20sylibr 216 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/) )
22 inss2 3652 . . . . . . . . 9  |-  ( A  i^i  ran  F )  C_ 
ran  F
23 frn 5733 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
245, 23syl 17 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ran 
F  C_  RR )
2522, 24syl5ss 3442 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( A  i^i  ran  F
)  C_  RR )
2625adantr 467 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  RR )
27 reldisj 3807 . . . . . . 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 17 . . . . . 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 214 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) )
30 imass2 5203 . . . . 5  |-  ( ( A  i^i  ran  F
)  C_  ( RR  \  { 0 } )  ->  ( `' F " ( A  i^i  ran  F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3129, 30syl 17 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3215, 31eqsstrd 3465 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
33 i1fima 22629 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol )
3433adantr 467 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  e.  dom  vol )
35 mblss 22478 . . . 4  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( `' F "
( RR  \  {
0 } ) ) 
C_  RR )
3634, 35syl 17 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  C_  RR )
37 mblvol 22477 . . . . 5  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  =  ( vol* `  ( `' F " ( RR 
\  { 0 } ) ) ) )
3834, 37syl 17 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  =  ( vol* `  ( `' F " ( RR  \  { 0 } ) ) ) )
39 isi1f 22625 . . . . . . 7  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
4039simprbi 466 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F : RR --> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
4140simp3d 1021 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
4241adantr 467 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR )
4338, 42eqeltrrd 2529 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " ( RR  \  {
0 } ) ) )  e.  RR )
44 ovolsscl 22432 . . 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 1267 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " A ) )  e.  RR )
464, 45eqeltrd 2528 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    \ cdif 3400    i^i cin 3402    C_ wss 3403   (/)c0 3730   {csn 3967   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   Fun wfun 5575   -->wf 5577   ` cfv 5581   Fincfn 7566   RRcr 9535   0cc0 9536   vol*covol 22406   volcvol 22408  MblFncmbf 22565   S.1citg1 22566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xadd 11407  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-xmet 18956  df-met 18957  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571
This theorem is referenced by:  i1fima2sn  22631  i1f0rn  22633  itg2addnclem  31986  itg2addnclem2  31987  ftc1anclem3  32012
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