Theorem isfsuppfi 7296

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
isfsuppfi	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))))

Distinct variable groups:   𝑤,𝑅   𝑤,𝑍

Allowed substitution hints:   𝑈(𝑤)   𝑉(𝑤)

Proof of Theorem isfsuppfi

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

Step	Hyp	Ref	Expression
1		funeq 5400	. . . 4 ⊢ (𝑟 = 𝑅 → (Fun 𝑟 ↔ Fun 𝑅))
2	1	adantr 276	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (Fun 𝑟 ↔ Fun 𝑅))
3		dmeq 4983	. . . . . . 7 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4	3	adantr 276	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → dom 𝑟 = dom 𝑅)
5	4	sseq2d 3278	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑤 ⊆ dom 𝑟 ↔ 𝑤 ⊆ dom 𝑅))
6		simpl 109	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → 𝑟 = 𝑅)
7	4	difeq1d 3346	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (dom 𝑟 ∖ 𝑤) = (dom 𝑅 ∖ 𝑤))
8	6, 7	imaeq12d 5130	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (𝑟 “ (dom 𝑟 ∖ 𝑤)) = (𝑅 “ (dom 𝑅 ∖ 𝑤)))
9		sneq 3720	. . . . . . 7 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
10	9	adantl 277	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
11	8, 10	sseq12d 3279	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧} ↔ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))
12	5, 11	3anbi23d 1356	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑟 ∧ (𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧}) ↔ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍})))
13	12	exbidv 1878	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → (∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑟 ∧ (𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧}) ↔ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍})))
14	2, 13	anbi12d 477	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = 𝑍) → ((Fun 𝑟 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑟 ∧ (𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧})) ↔ (Fun 𝑅 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))))
15		df-fsuppfi 7293	. 2 ⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑟 ∧ (𝑟 “ (dom 𝑟 ∖ 𝑤)) ⊆ {𝑧}))}
16	14, 15	brabga 4408	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ∖ cdif 3217   ⊆ wss 3220  {csn 3709   class class class wbr 4131  dom cdm 4776   “ cima 4779  Fun wfun 5374  Fincfn 7028   finSupp cfsupp 7292

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-fun 5382  df-fsuppfi 7293

This theorem is used by:  isfsuppfid  7297  funisfsuppdc  7298  fsuppfund  7299