Theorem fsuppimp 9281

Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)

Assertion

Ref	Expression
fsuppimp	⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

Proof of Theorem fsuppimp

Step	Hyp	Ref	Expression
1		relfsupp 9276	. . 3 ⊢ Rel finSupp
2	1	brrelex12i 5686	. 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
3		isfsupp 9278	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
4	3	biimpd 229	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
5	2, 4	mpcom 38	1 ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085  Fun wfun 6492  (class class class)co 7367   supp csupp 8110  Fincfn 8893   finSupp cfsupp 9274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-fsupp 9275

This theorem is referenced by:  fsuppimpd  9282  fsuppfund  9283  fsuppunfi  9301  fsuppunbi  9302  fsuppres  9306  fsuppco  9315  oemapvali  9605  mptnn0fsuppr  13961  gsumzres  19884  gsumzf1o  19887