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Theorem funisfsupp 7841
Description: The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
Assertion
Ref Expression
funisfsupp  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( R supp  Z
)  e.  Fin )
)

Proof of Theorem funisfsupp
StepHypRef Expression
1 isfsupp 7840 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
213adant1 1023 . 2  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) ) )
3 ibar 506 . . . 4  |-  ( Fun 
R  ->  ( ( R supp  Z )  e.  Fin  <->  ( Fun  R  /\  ( R supp 
Z )  e.  Fin ) ) )
43bicomd 204 . . 3  |-  ( Fun 
R  ->  ( ( Fun  R  /\  ( R supp 
Z )  e.  Fin ) 
<->  ( R supp  Z )  e.  Fin ) )
543ad2ant1 1026 . 2  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  (
( Fun  R  /\  ( R supp  Z )  e.  Fin )  <->  ( R supp  Z )  e.  Fin )
)
62, 5bitrd 256 1  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( R supp  Z
)  e.  Fin )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1872   class class class wbr 4366   Fun wfun 5538  (class class class)co 6249   supp csupp 6869   Fincfn 7524   finSupp cfsupp 7836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-rel 4803  df-cnv 4804  df-co 4805  df-iota 5508  df-fun 5546  df-fv 5552  df-ov 6252  df-fsupp 7837
This theorem is referenced by:  suppeqfsuppbi  7850  suppssfifsupp  7851  fsuppunbi  7857  0fsupp  7858  snopfsupp  7859  fsuppres  7861  resfsupp  7863  frnfsuppbi  7865  fsuppco  7868  sniffsupp  7876  cantnfp1lem1  8135  mptnn0fsupp  12159  dprdfadd  17596  lcomfsupp  18071  mplsubglem2  18603  ltbwe  18639  frlmbas  19260  frlmphllem  19280  frlmsslsp  19296  pmatcollpw2lem  19743  rrxmval  22301  eulerpartgbij  29157  pwfi2f1o  35867  lcoc0  39818
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