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Theorem fsuppimp 8442
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 8438 . . . 4 Rel finSupp
21brrelexi 5311 . . 3 (𝑅 finSupp 𝑍𝑅 ∈ V)
31brrelex2i 5312 . . 3 (𝑅 finSupp 𝑍𝑍 ∈ V)
42, 3jca 555 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
5 isfsupp 8440 . . 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 383  wcel 2135  Vcvv 3336   class class class wbr 4800  Fun wfun 6039  (class class class)co 6809   supp csupp 7459  Fincfn 8117   finSupp cfsupp 8436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-fsupp 8437
This theorem is referenced by:  fsuppimpd  8443  fsuppunfi  8456  fsuppunbi  8457  fsuppres  8461  fsuppco  8468  oemapvali  8750  mptnn0fsuppr  12989  gsumzres  18506  gsumzf1o  18509
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