Theorem isfsupp 9435

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 6598	. . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
2	1	adantr 480	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3		oveq12 7457	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
4	3	eleq1d 2829	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
5	2, 4	anbi12d 631	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6		df-fsupp 9432	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
7	5, 6	brabga 5553	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  Fun wfun 6567  (class class class)co 7448   supp csupp 8201  Fincfn 9003   finSupp cfsupp 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-fsupp 9432

This theorem is referenced by:  isfsuppd  9436  funisfsupp  9437  fsuppimp  9438  fdmfifsupp  9444  fczfsuppd  9455  fsuppmptif  9468  fsuppco2  9472  fsuppcor  9473  gsumzadd  19964  gsumpt  20004  gsum2dlem2  20013  gsum2d  20014  gsum2d2lem  20015  mhpmulcl  22176  rmfsupp2  33218  elrspunidl  33421  naddcnff  43324  rmfsupp  48099  mndpfsupp  48101  scmfsupp  48103  mptcfsupp  48105