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Theorem fsuppimpd 8557
Description: A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)
Hypothesis
Ref Expression
fsuppimpd.f (𝜑𝐹 finSupp 𝑍)
Assertion
Ref Expression
fsuppimpd (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Proof of Theorem fsuppimpd
StepHypRef Expression
1 fsuppimpd.f . 2 (𝜑𝐹 finSupp 𝑍)
2 fsuppimp 8556 . . 3 (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
32simprd 491 . 2 (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin)
41, 3syl 17 1 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164   class class class wbr 4875  Fun wfun 6121  (class class class)co 6910   supp csupp 7564  Fincfn 8228   finSupp cfsupp 8550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  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 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-fsupp 8551
This theorem is referenced by:  fsuppsssupp  8566  fsuppxpfi  8567  fsuppun  8569  resfsupp  8577  fsuppmptif  8580  fsuppco  8582  fsuppco2  8583  fsuppcor  8584  cantnfcl  8848  cantnfp1lem1  8859  fsuppmapnn0fiublem  13091  fsuppmapnn0fiub  13092  fsuppmapnn0ub  13096  gsumzcl  18672  gsumcl  18676  gsumzadd  18682  gsumzmhm  18697  gsumzoppg  18704  gsum2dlem1  18729  gsum2dlem2  18730  gsum2d  18731  gsumdixp  18970  lcomfsupp  19266  mptscmfsupp0  19291  mplcoe1  19833  mplbas2  19838  psrbagev1  19877  evlslem2  19879  evlslem6  19880  regsumsupp  20336  frlmphllem  20493  uvcresum  20506  frlmsslsp  20509  frlmup1  20511  tsmsgsum  22319  rrxcph  23567  rrxfsupp  23577  mdegldg  24232  mdegcl  24235  plypf1  24374  rmfsupp  43016  mndpfsupp  43018  scmfsupp  43020  lincresunit2  43128
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