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Theorem fsuppfund 9408
Description: A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
Hypothesis
Ref Expression
fsuppfund.1 (𝜑𝐹 finSupp 𝑍)
Assertion
Ref Expression
fsuppfund (𝜑 → Fun 𝐹)

Proof of Theorem fsuppfund
StepHypRef Expression
1 fsuppfund.1 . 2 (𝜑𝐹 finSupp 𝑍)
2 fsuppimp 9406 . . 3 (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
32simpld 494 . 2 (𝐹 finSupp 𝑍 → Fun 𝐹)
41, 3syl 17 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5148  Fun wfun 6557  (class class class)co 7431   supp csupp 8184  Fincfn 8984   finSupp cfsupp 9399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-fsupp 9400
This theorem is referenced by:  fsuppss  9421
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