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Theorem fsuppimp 8225
Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
Assertion
Ref Expression
fsuppimp (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

Proof of Theorem fsuppimp
StepHypRef Expression
1 relfsupp 8221 . . . 4 Rel finSupp
21brrelexi 5118 . . 3 (𝑅 finSupp 𝑍𝑅 ∈ V)
31brrelex2i 5119 . . 3 (𝑅 finSupp 𝑍𝑍 ∈ V)
42, 3jca 554 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
5 isfsupp 8223 . . 3 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
65biimpd 219 . 2 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
74, 6mpcom 38 1 (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186   class class class wbr 4613  Fun wfun 5841  (class class class)co 6604   supp csupp 7240  Fincfn 7899   finSupp cfsupp 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-fsupp 8220
This theorem is referenced by:  fsuppimpd  8226  fsuppunfi  8239  fsuppunbi  8240  fsuppres  8244  fsuppco  8251  oemapvali  8525  mptnn0fsuppr  12739  gsumzres  18231  gsumzf1o  18234
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