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Theorem isfsupp 7891
Description: The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
Assertion
Ref Expression
isfsupp  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )

Proof of Theorem isfsupp
Dummy variables  r 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funeq 5618 . . . 4  |-  ( r  =  R  ->  ( Fun  r  <->  Fun  R ) )
21adantr 467 . . 3  |-  ( ( r  =  R  /\  z  =  Z )  ->  ( Fun  r  <->  Fun  R ) )
3 oveq12 6312 . . . 4  |-  ( ( r  =  R  /\  z  =  Z )  ->  ( r supp  z )  =  ( R supp  Z
) )
43eleq1d 2492 . . 3  |-  ( ( r  =  R  /\  z  =  Z )  ->  ( ( r supp  z
)  e.  Fin  <->  ( R supp  Z )  e.  Fin )
)
52, 4anbi12d 716 . 2  |-  ( ( r  =  R  /\  z  =  Z )  ->  ( ( Fun  r  /\  ( r supp  z )  e.  Fin )  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
6 df-fsupp 7888 . 2  |- finSupp  =  { <. r ,  z >.  |  ( Fun  r  /\  ( r supp  z )  e.  Fin ) }
75, 6brabga 4732 1  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   class class class wbr 4421   Fun wfun 5593  (class class class)co 6303   supp csupp 6923   Fincfn 7575   finSupp cfsupp 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-rel 4858  df-cnv 4859  df-co 4860  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-fsupp 7888
This theorem is referenced by:  funisfsupp  7892  fsuppimp  7893  fdmfifsupp  7897  fczfsuppd  7905  fsuppmptif  7917  fsuppco2  7920  fsuppcor  7921  gsumzadd  17548  gsumpt  17587  gsum2dlem2  17596  gsum2d  17597  gsum2d2lem  17598  rmfsupp  39465  mndpfsupp  39467  scmfsupp  39469  mptcfsupp  39471
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