Theorem fsuppfund 41522

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 9363	. . 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 5138  Fun wfun 6527  (class class class)co 7401   supp csupp 8140  Fincfn 8934   finSupp cfsupp 9356

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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

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 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-fsupp 9357

This theorem is referenced by:  fsuppss  41524