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Theorem isfsupp 9367
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 ((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))

Proof of Theorem isfsupp
Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funeq 6567 . . . 4 (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
21adantr 479 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3 oveq12 7420 . . . 4 ((𝑟 = 𝑅𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
43eleq1d 2816 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
52, 4anbi12d 629 . 2 ((𝑟 = 𝑅𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6 df-fsupp 9364 . 2 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
75, 6brabga 5533 1 ((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104   class class class wbr 5147  Fun wfun 6536  (class class class)co 7411   supp csupp 8148  Fincfn 8941   finSupp cfsupp 9363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-fsupp 9364
This theorem is referenced by:  isfsuppd  9368  funisfsupp  9369  fsuppimp  9370  fdmfifsupp  9375  fczfsuppd  9383  fsuppmptif  9396  fsuppco2  9400  fsuppcor  9401  gsumzadd  19831  gsumpt  19871  gsum2dlem2  19880  gsum2d  19881  gsum2d2lem  19882  mhpmulcl  21911  rmfsupp2  32657  elrspunidl  32820  naddcnff  42414  rmfsupp  47138  mndpfsupp  47140  scmfsupp  47142  mptcfsupp  47144
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