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Theorem fsuppfund 9394
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 9392 . . 3 (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
32simpld 493 . 2 (𝐹 finSupp 𝑍 → Fun 𝐹)
41, 3syl 17 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5143  Fun wfun 6537  (class class class)co 7416   supp csupp 8163  Fincfn 8962   finSupp cfsupp 9385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-fsupp 9386
This theorem is referenced by:  fsuppss  9406
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