MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relfsupp Structured version   Visualization version   GIF version

Theorem relfsupp 8819
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.
StepHypRef Expression
1 df-fsupp 8818 . 2 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
21relopabi 5658 1 Rel finSupp
Colors of variables: wff setvar class
Syntax hints:  wa 399  wcel 2111  Rel wrel 5524  Fun wfun 6318  (class class class)co 7135   supp csupp 7813  Fincfn 8492   finSupp cfsupp 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526  df-fsupp 8818
This theorem is referenced by:  relprcnfsupp  8820  fsuppimp  8823  suppeqfsuppbi  8831  fsuppsssupp  8833  fsuppunbi  8838  funsnfsupp  8841  wemapso2  9001  gsumhashmul  30741
  Copyright terms: Public domain W3C validator