Theorem relfsupp 8829

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 8828	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
2	1	relopabi 5688	1 ⊢ Rel finSupp

Syntax hints:   ∧ wa 398   ∈ wcel 2110  Rel wrel 5554  Fun wfun 6343  (class class class)co 7150   supp csupp 7824  Fincfn 8503   finSupp cfsupp 8827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121  df-xp 5555  df-rel 5556  df-fsupp 8828

This theorem is referenced by:  relprcnfsupp  8830  fsuppimp  8833  suppeqfsuppbi  8841  fsuppsssupp  8843  fsuppunbi  8848  funsnfsupp  8851  wemapso2  9011