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Theorem funisfsupp 7826
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 7825 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
213adant1 1012 . 2  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) ) )
3 ibar 502 . . . 4  |-  ( Fun 
R  ->  ( ( R supp  Z )  e.  Fin  <->  ( Fun  R  /\  ( R supp 
Z )  e.  Fin ) ) )
43bicomd 201 . . 3  |-  ( Fun 
R  ->  ( ( Fun  R  /\  ( R supp 
Z )  e.  Fin ) 
<->  ( R supp  Z )  e.  Fin ) )
543ad2ant1 1015 . 2  |-  ( ( Fun  R  /\  R  e.  V  /\  Z  e.  W )  ->  (
( Fun  R  /\  ( R supp  Z )  e.  Fin )  <->  ( R supp  Z )  e.  Fin )
)
62, 5bitrd 253 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 184    /\ wa 367    /\ w3a 971    e. wcel 1823   class class class wbr 4439   Fun wfun 5564  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-rel 4995  df-cnv 4996  df-co 4997  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-fsupp 7822
This theorem is referenced by:  suppeqfsuppbi  7835  suppssfifsupp  7836  fsuppunbi  7842  0fsupp  7843  snopfsupp  7844  fsuppres  7846  resfsupp  7848  frnfsuppbi  7850  fsuppco  7853  sniffsupp  7861  cantnffvalOLD  8073  cantnfp1lem1  8088  mptnn0fsupp  12088  dprdvalOLD  17234  dprdfadd  17258  lcomfsupp  17748  mplsubglem2  18295  ltbwe  18335  frlmbas  18962  frlmphllem  18985  frlmsslsp  19001  pmatcollpw2lem  19448  rrxmval  22001  eulerpartgbij  28578  pwfi2f1o  31286  lcoc0  33296
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