Theorem isfsupp 9280

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 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (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.

Step	Hyp	Ref	Expression
1		funeq 6520	. . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
2	1	adantr 480	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3		oveq12 7377	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
4	3	eleq1d 2822	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
5	2, 4	anbi12d 633	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6		df-fsupp 9277	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
7	5, 6	brabga 5490	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  Fun wfun 6494  (class class class)co 7368   supp csupp 8112  Fincfn 8895   finSupp cfsupp 9276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-fsupp 9277

This theorem is referenced by:  isfsuppd  9281  funisfsupp  9282  fsuppimp  9283  fdmfifsupp  9290  fczfsuppd  9301  fsuppmptif  9314  fsuppco2  9318  fsuppcor  9319  mndpfsupp  18704  gsumzadd  19863  gsumpt  19903  gsum2dlem2  19912  gsum2d  19913  gsum2d2lem  19914  mhpmulcl  22104  rmfsupp2  33332  elrspunidl  33521  psrbasfsupp  33705  naddcnff  43719  rmfsupp  48733  scmfsupp  48735  mptcfsupp  48737