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Theorem fsuppimp 8437
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 8433 . . . 4 Rel finSupp
21brrelexi 5298 . . 3 (𝑅 finSupp 𝑍𝑅 ∈ V)
31brrelex2i 5299 . . 3 (𝑅 finSupp 𝑍𝑍 ∈ V)
42, 3jca 501 . 2 (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
5 isfsupp 8435 . . 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 382  wcel 2145  Vcvv 3351   class class class wbr 4786  Fun wfun 6025  (class class class)co 6793   supp csupp 7446  Fincfn 8109   finSupp cfsupp 8431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-fsupp 8432
This theorem is referenced by:  fsuppimpd  8438  fsuppunfi  8451  fsuppunbi  8452  fsuppres  8456  fsuppco  8463  oemapvali  8745  mptnn0fsuppr  13006  gsumzres  18517  gsumzf1o  18520
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