Theorem fsuppimp 9338

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 9333	. . 3 ⊢ Rel finSupp
2	1	brrelex12i 5710	. 2 ⊢ (𝑅 finSupp 𝑍 → (𝑅 ∈ V ∧ 𝑍 ∈ V))
3		isfsupp 9335	. . 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 39	1 ⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  Vcvv 3450   class class class wbr 5103  Fun wfun 6527  (class class class)co 7413   supp csupp 8158  Fincfn 8952   finSupp cfsupp 9331

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-fsupp 9332

This theorem is used by:  fsuppimpd  9339  fsuppfund  9340  fsuppunfi  9358  fsuppunbi  9359  fsuppres  9363  fsuppco  9372  oemapvali  9663  mptnn0fsuppr  14063  gsumzres  20036  gsumzf1o  20039