Theorem isfsuppfid 7297

Description: Deduction form of isfsuppfi 7296. (Contributed by SN, 29-Jul-2024.) (Revised by Jim Kingdon, 16-Oct-2026.)

Hypotheses

Ref	Expression
isfsuppfid.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
isfsuppfid.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
isfsuppfid.1	⊢ (𝜑 → Fun 𝑅)
isfsuppfid.w	⊢ (𝜑 → 𝑊 ∈ Fin)
isfsuppfid.wss	⊢ (𝜑 → 𝑊 ⊆ dom 𝑅)
isfsuppfid.wz	⊢ (𝜑 → (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍})

Assertion

Ref	Expression
isfsuppfid	⊢ (𝜑 → 𝑅 finSupp 𝑍)

Proof of Theorem isfsuppfid

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfsuppfid.1	. 2 ⊢ (𝜑 → Fun 𝑅)
2		isfsuppfid.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ Fin)
3		isfsuppfid.wss	. . 3 ⊢ (𝜑 → 𝑊 ⊆ dom 𝑅)
4		isfsuppfid.wz	. . 3 ⊢ (𝜑 → (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍})
5		eleq1 2301	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Fin ↔ 𝑊 ∈ Fin))
6		sseq1 3271	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤 ⊆ dom 𝑅 ↔ 𝑊 ⊆ dom 𝑅))
7		difeq2 3341	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dom 𝑅 ∖ 𝑤) = (dom 𝑅 ∖ 𝑊))
8	7	imaeq2d 5129	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑅 “ (dom 𝑅 ∖ 𝑤)) = (𝑅 “ (dom 𝑅 ∖ 𝑊)))
9	8	sseq1d 3277	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍} ↔ (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍}))
10	5, 6, 9	3anbi123d 1353	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}) ↔ (𝑊 ∈ Fin ∧ 𝑊 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍})))
11	10	spcegv 2913	. . . 4 ⊢ (𝑊 ∈ Fin → ((𝑊 ∈ Fin ∧ 𝑊 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍}) → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍})))
12	2, 11	syl 14	. . 3 ⊢ (𝜑 → ((𝑊 ∈ Fin ∧ 𝑊 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑊)) ⊆ {𝑍}) → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍})))
13	2, 3, 4, 12	mp3and 1381	. 2 ⊢ (𝜑 → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))
14		isfsuppfid.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
15		isfsuppfid.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
16		isfsuppfi 7296	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))))
17	14, 15, 16	syl2anc 415	. 2 ⊢ (𝜑 → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝑅 ∧ (𝑅 “ (dom 𝑅 ∖ 𝑤)) ⊆ {𝑍}))))
18	1, 13, 17	mpbir2and 957	1 ⊢ (𝜑 → 𝑅 finSupp 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-in1 623  ax-in2 624  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-ral 2533  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:  funisfsuppdc  7298  fdmfifsupp  7300  fczfsuppd  7303  0fsupp  7304  snopfsupp  7305