Theorem fsuppfund 9275

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 9273	. . 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 2113   class class class wbr 5098  Fun wfun 6486  (class class class)co 7358   supp csupp 8102  Fincfn 8885   finSupp cfsupp 9266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-fsupp 9267

This theorem is referenced by:  fsuppss  9288