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Theorem funisfsupp 9274

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 9272	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
2	1	3adant1 1131	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
3		ibar 528	. . . 4 ⊢ (Fun 𝑅 → ((𝑅 supp 𝑍) ∈ Fin ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
4	3	bicomd 223	. . 3 ⊢ (Fun 𝑅 → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
5	4	3ad2ant1 1134	. 2 ⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
6	2, 5	bitrd 279	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 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 Fun wfun 6487 (class class class)co 7361 supp csupp 8104 Fincfn 8887 finSupp cfsupp 9268

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5232 ax-pr 5371

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-rel 5632 df-cnv 5633 df-co 5634 df-iota 6449 df-fun 6495 df-fv 6501 df-ov 7364 df-fsupp 9269

This theorem is referenced by: fidmfisupp 9279 finnzfsuppd 9280 suppeqfsuppbi 9286 suppssfifsupp 9287 fsuppunbi 9296 0fsupp 9297 snopfsupp 9298 fsuppres 9300 resfsupp 9303 ffsuppbi 9305 sniffsupp 9307 fsuppco 9309 cantnfp1lem1 9593 fcdmnn0fsuppg 12491 mptnn0fsupp 13953 dprdfadd 19991 lcomfsupp 20891 frlmbas 21748 frlmphllem 21773 frlmsslsp 21789 mplsubglem2 21992 ltbwe 22035 pmatcollpw2lem 22755 rrxmval 25385 offinsupp1 32817 elrspunidl 33506 eulerpartgbij 34535 pwfi2f1o 43545 cantnfub 43770 lcoc0 48913

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