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Theorem fsuppimp 8550
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 8546 . . . 4 Rel finSupp
21brrelex1i 5393 . . 3 (𝑅 finSupp 𝑍𝑅 ∈ V)
31brrelex2i 5394 . . 3 (𝑅 finSupp 𝑍𝑍 ∈ V)
42, 3jca 509 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
5 isfsupp 8548 . . 3 ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
65biimpd 221 . 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 386  wcel 2166  Vcvv 3414   class class class wbr 4873  Fun wfun 6117  (class class class)co 6905   supp csupp 7559  Fincfn 8222   finSupp cfsupp 8544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-fsupp 8545
This theorem is referenced by:  fsuppimpd  8551  fsuppunfi  8564  fsuppunbi  8565  fsuppres  8569  fsuppco  8576  oemapvali  8858  mptnn0fsuppr  13093  gsumzres  18663  gsumzf1o  18666
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