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Theorem isfsupp 9316
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 6526 . . . 4 (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
21adantr 482 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3 oveq12 7371 . . . 4 ((𝑟 = 𝑅𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
43eleq1d 2823 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
52, 4anbi12d 632 . 2 ((𝑟 = 𝑅𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6 df-fsupp 9313 . 2 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
75, 6brabga 5496 1 ((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5110  Fun wfun 6495  (class class class)co 7362   supp csupp 8097  Fincfn 8890   finSupp cfsupp 9312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-rel 5645  df-cnv 5646  df-co 5647  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-fsupp 9313
This theorem is referenced by:  funisfsupp  9317  fsuppimp  9318  fdmfifsupp  9322  fczfsuppd  9330  fsuppmptif  9342  fsuppco2  9346  fsuppcor  9347  gsumzadd  19706  gsumpt  19746  gsum2dlem2  19755  gsum2d  19756  gsum2d2lem  19757  mhpmulcl  21555  rmfsupp2  32115  elrspunidl  32243  isfsuppd  40695  naddcnff  41707  rmfsupp  46524  mndpfsupp  46526  scmfsupp  46528  mptcfsupp  46530
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