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Theorem isfsupp 7853
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

Proof of Theorem isfsupp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funeq 5612 . . . 4
21adantr 465 . . 3
3 oveq12 6305 . . . 4
43eleq1d 2526 . . 3
52, 4anbi12d 710 . 2
6 df-fsupp 7850 . 2
75, 6brabga 4766 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   class class class wbr 4452  Funwfun 5587  (class class class)co 6296   csupp 6918   cfn 7536   cfsupp 7849
This theorem is referenced by:  funisfsupp  7854  fsuppimp  7855  fdmfifsupp  7859  fczfsuppd  7867  fsuppmptif  7879  fsuppco2  7882  fsuppcor  7883  gsumzadd  16935  gsumpt  16988  gsum2dlem2  16998  gsum2d  16999  gsum2d2lem  17001  mplvalOLD  18085  rmfsupp  32967  mndpfsupp  32969  scmfsupp  32971  mptcfsupp  32973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-rel 5011  df-cnv 5012  df-co 5013  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-fsupp 7850
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