Theorem relfsupp 9389

Description: The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)

Assertion

Ref	Expression
relfsupp	⊢ Rel finSupp

Proof of Theorem relfsupp

Dummy variables 𝑧 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fsupp 9388	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5822	1 ⊢ Rel finSupp

Syntax hints:   ∧ wa 394   ∈ wcel 2098  Rel wrel 5683  Fun wfun 6543  (class class class)co 7419   supp csupp 8165  Fincfn 8964   finSupp cfsupp 9387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-ss 3961  df-opab 5212  df-xp 5684  df-rel 5685  df-fsupp 9388

This theorem is referenced by:  relprcnfsupp  9390  fsuppimp  9394  suppeqfsuppbi  9404  fsuppsssupp  9406  fsuppss  9408  fsuppssov1  9409  fsuppunbi  9414  funsnfsupp  9417  wemapso2  9578  gsumhashmul  32860