Theorem relfsupp 9330

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 9329	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5809	1 ⊢ Rel finSupp

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2146  Rel wrel 5668  Fun wfun 6534  (class class class)co 7419   supp csupp 8162  Fincfn 8949   finSupp cfsupp 9328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-opab 5176  df-xp 5669  df-rel 5670  df-fsupp 9329

This theorem is used by:  relprcnfsupp  9331  fsuppimp  9335  suppeqfsuppbi  9346  fsuppsssupp  9348  fsuppss  9350  fsuppssov1  9351  fsuppunbi  9356  funsnfsupp  9359  wemapso2  9522  gsumhashmul  33451