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Theorem funisfsupp 9373

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 9371	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
2	1	3adant1 1129	. 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 222	. . 3 ⊢ (Fun 𝑅 → ((Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin) ↔ (𝑅 supp 𝑍) ∈ Fin))
5	4	3ad2ant1 1132	. 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 205 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2105 class class class wbr 5148 Fun wfun 6537 (class class class)co 7412 supp csupp 8151 Fincfn 8945 finSupp cfsupp 9367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-rel 5683 df-cnv 5684 df-co 5685 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-fsupp 9368

This theorem is referenced by: fidmfisupp 9377 suppeqfsuppbi 9383 suppssfifsupp 9384 fsuppunbi 9390 0fsupp 9391 snopfsupp 9392 fsuppres 9394 resfsupp 9397 ffsuppbi 9399 sniffsupp 9401 fsuppco 9403 cantnfp1lem1 9679 fcdmnn0fsuppg 12538 mptnn0fsupp 13969 dprdfadd 19938 lcomfsupp 20744 frlmbas 21620 frlmphllem 21645 frlmsslsp 21661 mplsubglem2 21871 ltbwe 21910 pmatcollpw2lem 22599 rrxmval 25253 offinsupp1 32385 elrspunidl 32986 eulerpartgbij 33835 pwfi2f1o 42301 cantnfub 42534 finnzfsuppd 43424 lcoc0 47265

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