Theorem relfsupp 9365

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 9364	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5813	1 ⊢ Rel finSupp

Syntax hints:   ∧ wa 395   ∈ wcel 2098  Rel wrel 5674  Fun wfun 6531  (class class class)co 7405   supp csupp 8146  Fincfn 8941   finSupp cfsupp 9363

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-opab 5204  df-xp 5675  df-rel 5676  df-fsupp 9364

This theorem is referenced by:  relprcnfsupp  9366  fsuppimp  9370  suppeqfsuppbi  9379  fsuppsssupp  9381  fsuppunbi  9386  funsnfsupp  9389  wemapso2  9550  gsumhashmul  32714  fsuppss  41626