Theorem fsuppfund 9260

Description: A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)

Hypothesis

Ref	Expression
fsuppfund.1	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppfund	⊢ (𝜑 → Fun 𝐹)

Proof of Theorem fsuppfund

Step	Hyp	Ref	Expression
1		fsuppfund.1	. 2 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
2		fsuppimp 9258	. . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
3	2	simpld 494	. 2 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹)
4	1, 3	syl 17	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5092  Fun wfun 6476  (class class class)co 7349   supp csupp 8093  Fincfn 8872   finSupp cfsupp 9251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-fsupp 9252

This theorem is referenced by:  fsuppss  9273