Theorem relfsupp 9360

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 9359	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabiv 5798	1 ⊢ Rel finSupp

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2144  Rel wrel 5655  Fun wfun 6535  (class class class)co 7422   supp csupp 8181  Fincfn 8978   finSupp cfsupp 9358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-opab 5167  df-xp 5656  df-rel 5657  df-fsupp 9359

This theorem is used by:  relprcnfsupp  9361  fsuppimp  9365  suppeqfsuppbi  9376  fsuppsssupp  9378  fsuppss  9380  fsuppssov1  9381  fsuppunbi  9386  funsnfsupp  9389  wemapso2  9552  gsumhashmul  33676