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Theorem funisfsupp 8837

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 8836	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
2	1	3adant1 1126	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
3		ibar 531	. . . 4 ⊢ (Fun 𝑅 → ((𝑅 supp 𝑍) ∈ Fin ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
4	3	bicomd 225	. . 3 ⊢ (Fun 𝑅 → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
5	4	3ad2ant1 1129	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
6	2, 5	bitrd 281	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 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5065 Fun wfun 6348 (class class class)co 7155 supp csupp 7829 Fincfn 8508 finSupp cfsupp 8832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-rel 5561 df-cnv 5562 df-co 5563 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-fsupp 8833

This theorem is referenced by: suppeqfsuppbi 8846 suppssfifsupp 8847 fsuppunbi 8853 0fsupp 8854 snopfsupp 8855 fsuppres 8857 resfsupp 8859 frnfsuppbi 8861 fsuppco 8864 sniffsupp 8872 cantnfp1lem1 9140 mptnn0fsupp 13364 dprdfadd 19141 lcomfsupp 19673 mplsubglem2 20215 ltbwe 20252 frlmbas 20898 frlmphllem 20923 frlmsslsp 20939 pmatcollpw2lem 21384 rrxmval 24007 offinsupp1 30462 eulerpartgbij 31630 pwfi2f1o 39694 fidmfisupp 41460 lcoc0 44476

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