Theorem isfsupp 9325

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

Assertion

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

Proof of Theorem isfsupp

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funeq 6557	. . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
2	1	adantr 485	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3		oveq12 7420	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 supp 𝑧) = (𝑅 supp 𝑍))
4	3	eleq1d 2854	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 supp 𝑧) ∈ Fin ↔ (𝑅 supp 𝑍) ∈ Fin))
5	2, 4	anbi12d 643	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin) ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6		df-fsupp 9322	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
7	5, 6	brabga 5519	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  Fun wfun 6531  (class class class)co 7411   supp csupp 8156  Fincfn 8943   finSupp cfsupp 9321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-rel 5669  df-cnv 5670  df-co 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-fsupp 9322

This theorem is referenced by:  isfsuppd  9326  funisfsupp  9327  fsuppimp  9328  fdmfifsupp  9335  fsuppmptif  9359  fsuppco2  9363  fsuppcor  9364  mndpfsupp  18825  gsumzadd  19992  gsumpt  20032  gsum2dlem2  20041  gsum2d  20042  gsum2d2lem  20043  mhpmulcl  22281  rmfsupp2  33498  elrspunidl  33680  psrbasfsupp  33846  naddcnff  44015  rmfsupp  49072  scmfsupp  49074  mptcfsupp  49076