Theorem fsuppfund 41624

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 9370	. . 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 2098   class class class wbr 5141  Fun wfun 6531  (class class class)co 7405   supp csupp 8146  Fincfn 8941   finSupp cfsupp 9363

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-fsupp 9364

This theorem is referenced by:  fsuppss  41626