Theorem relfsupp 7294

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 𝑤 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fsuppfi 7293	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑟 ∧ (𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧}))}
2	1	relopabi 4907	1 ⊢ Rel finSupp

This proof depends on syntax axioms:   ∧ wa 104   ∧ w3a 1009  ∃wex 1545   ∈ wcel 2209   ∖ cdif 3217   ⊆ wss 3220  {csn 3709  dom cdm 4776   “ cima 4779  Rel wrel 4781  Fun wfun 5374  Fincfn 7028   finSupp cfsupp 7292

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4194  df-xp 4782  df-rel 4783  df-fsuppfi 7293

This theorem is used by:  relprcnfsupp  7295  fsuppfund  7299  fsuppfiss  7300