Theorem fsuppimp 8164

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 8160	. . . 4 ⊢ Rel finSupp
2	1	brrelexi 5082	. . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑅 ∈ V)
3	1	brrelex2i 5083	. . 3 ⊢ (𝑅 finSupp 𝑍 → 𝑍 ∈ V)
4	2, 3	jca 553	. 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
5		isfsupp 8162	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6	5	biimpd 218	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑍 ∈ V) → (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
7	4, 6	mpcom 37	1 ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  Fun wfun 5798  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-fsupp 8159

This theorem is referenced by:  fsuppimpd  8165  fsuppunfi  8178  fsuppunbi  8179  fsuppres  8183  fsuppco  8190  oemapvali  8464  mptnn0fsuppr  12661  gsumzres  18133  gsumzf1o  18136