Theorem relfsupp 9333

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 9332	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5801	1 ⊢ Rel finSupp

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2145  Rel wrel 5660  Fun wfun 6527  (class class class)co 7413   supp csupp 8158  Fincfn 8952   finSupp cfsupp 9331

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 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3916  df-opab 5168  df-xp 5661  df-rel 5662  df-fsupp 9332

This theorem is used by:  relprcnfsupp  9334  fsuppimp  9338  suppeqfsuppbi  9349  fsuppsssupp  9351  fsuppss  9353  fsuppssov1  9354  fsuppunbi  9359  funsnfsupp  9362  wemapso2  9525  gsumhashmul  33507