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Theorem isfsupp 8485
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 6087 . . . 4 (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
21adantr 472 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3 oveq12 6850 . . . 4 ((𝑟 = 𝑅𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
43eleq1d 2828 . . 3 ((𝑟 = 𝑅𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
52, 4anbi12d 624 . 2 ((𝑟 = 𝑅𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6 df-fsupp 8482 . 2 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
75, 6brabga 5149 1 ((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155   class class class wbr 4808  Fun wfun 6061  (class class class)co 6841   supp csupp 7496  Fincfn 8159   finSupp cfsupp 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-rel 5283  df-cnv 5284  df-co 5285  df-iota 6030  df-fun 6069  df-fv 6075  df-ov 6844  df-fsupp 8482
This theorem is referenced by:  funisfsupp  8486  fsuppimp  8487  fdmfifsupp  8491  fczfsuppd  8499  fsuppmptif  8511  fsuppco2  8514  fsuppcor  8515  gsumzadd  18587  gsumpt  18626  gsum2dlem2  18635  gsum2d  18636  gsum2d2lem  18637  rmfsupp  42756  mndpfsupp  42758  scmfsupp  42760  mptcfsupp  42762
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