Theorem relfsupp 9397

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 9396	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5826	1 ⊢ Rel finSupp

Syntax hints:   ∧ wa 394   ∈ wcel 2098  Rel wrel 5687  Fun wfun 6547  (class class class)co 7426   supp csupp 8173  Fincfn 8972   finSupp cfsupp 9395

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 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-xp 5688  df-rel 5689  df-fsupp 9396

This theorem is referenced by:  relprcnfsupp  9398  fsuppimp  9402  suppeqfsuppbi  9412  fsuppsssupp  9414  fsuppss  9416  fsuppssov1  9417  fsuppunbi  9422  funsnfsupp  9425  wemapso2  9586  gsumhashmul  32799