Theorem fsuppfund 9274

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 9272	. . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
3	2	simpld 495	. 2 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹)
4	1, 3	syl 17	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2119   class class class wbr 5073  Fun wfun 6480  (class class class)co 7357   supp csupp 8101  Fincfn 8884   finSupp cfsupp 9265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7360  df-fsupp 9266

This theorem is referenced by:  fsuppss  9287