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Theorem fsuppfund 9411
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 9409 . . 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 2107   class class class wbr 5142  Fun wfun 6554  (class class class)co 7432   supp csupp 8186  Fincfn 8986   finSupp cfsupp 9402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-fsupp 9403
This theorem is referenced by:  fsuppss  9424
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