Theorem fsuppimp 8823

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 8819	. . 3 ⊢ Rel finSupp
2	1	brrelex12i 5571	. 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
3		isfsupp 8821	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
4	3	biimpd 232	. 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 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030  Fun wfun 6318  (class class class)co 7135   supp csupp 7813  Fincfn 8492   finSupp cfsupp 8817

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-fsupp 8818

This theorem is referenced by:  fsuppimpd  8824  fsuppunfi  8837  fsuppunbi  8838  fsuppres  8842  fsuppco  8849  oemapvali  9131  mptnn0fsuppr  13362  gsumzres  19022  gsumzf1o  19025