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Theorem fsuppimp 8827
 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 8823 . . 3 Rel finSupp
21brrelex12i 5584 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
3 isfsupp 8825 . . 3 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
43biimpd 232 . 2 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
52, 4mpcom 38 1 (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  Vcvv 3469   class class class wbr 5042  Fun wfun 6328  (class class class)co 7140   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-fsupp 8822 This theorem is referenced by:  fsuppimpd  8828  fsuppunfi  8841  fsuppunbi  8842  fsuppres  8846  fsuppco  8853  oemapvali  9135  mptnn0fsuppr  13362  gsumzres  19020  gsumzf1o  19023
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