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Theorem funisfsupp 9391

Description: The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)

**Assertion**
Ref	Expression
funisfsupp	⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin))

**Proof of Theorem funisfsupp**
Step	Hyp	Ref	Expression
1		isfsupp 9389	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
2	1	3adant1 1127	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
3		ibar 527	. . . 4 ⊢ (Fun 𝑅 → ((𝑅 supp 𝑍) ∈ Fin ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
4	3	bicomd 222	. . 3 ⊢ (Fun 𝑅 → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
5	4	3ad2ant1 1130	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
6	2, 5	bitrd 278	1 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5143 Fun wfun 6537 (class class class)co 7416 supp csupp 8163 Fincfn 8962 finSupp cfsupp 9385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-rel 5679 df-cnv 5680 df-co 5681 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-fsupp 9386

This theorem is referenced by: fidmfisupp 9396 suppeqfsuppbi 9402 suppssfifsupp 9403 fsuppunbi 9412 0fsupp 9413 snopfsupp 9414 fsuppres 9416 resfsupp 9419 ffsuppbi 9421 sniffsupp 9423 fsuppco 9425 cantnfp1lem1 9701 fcdmnn0fsuppg 12561 mptnn0fsupp 13994 dprdfadd 19981 lcomfsupp 20789 frlmbas 21693 frlmphllem 21718 frlmsslsp 21734 mplsubglem2 21950 ltbwe 21989 pmatcollpw2lem 22697 rrxmval 25351 offinsupp1 32554 elrspunidl 33193 eulerpartgbij 34049 pwfi2f1o 42585 cantnfub 42815 finnzfsuppd 43704 lcoc0 47602

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