Theorem fsuppimpd 7283

Description: A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)

Hypothesis

Ref	Expression
fsuppimpd.f	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppimpd	⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Proof of Theorem fsuppimpd

Step	Hyp	Ref	Expression
1		fsuppimpd.f	. 2 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
2		fsuppimp 7282	. . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
3	2	simprd 114	. 2 ⊢ (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin)
4	1, 3	syl 14	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Syntax hints:   → wi 4   ∈ wcel 2209   class class class wbr 4125  Fun wfun 5366  (class class class)co 6075   supp csupp 6465  Fincfn 7012   finSupp cfsupp 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-fsupp 7276

This theorem is referenced by:  fsuppxpfi  7286  fsuppcorn  7291