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Theorem fsuppimp 9412
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 9407 . . 3 Rel finSupp
21brrelex12i 5737 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
3 isfsupp 9409 . . 3 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
43biimpd 228 . 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 394  wcel 2099  Vcvv 3462   class class class wbr 5153  Fun wfun 6548  (class class class)co 7424   supp csupp 8174  Fincfn 8974   finSupp cfsupp 9405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-fsupp 9406
This theorem is referenced by:  fsuppimpd  9413  fsuppfund  9414  fsuppunfi  9431  fsuppunbi  9432  fsuppres  9436  fsuppco  9445  oemapvali  9727  mptnn0fsuppr  14019  gsumzres  19907  gsumzf1o  19910
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