Theorem isfsuppf 7298

Description: Showing that a function is finitely supported. (Contributed by Jim Kingdon, 2-Nov-2026.)

Hypotheses

Ref	Expression
isfsuppf.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
isfsuppf.fex	⊢ (𝜑 → 𝐹 ∈ 𝑉)
isfsuppf.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
isfsuppf.w	⊢ (𝜑 → 𝑊 ∈ Fin)
isfsuppf.wss	⊢ (𝜑 → 𝑊 ⊆ 𝐴)
isfupppf.k	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)

Assertion

Ref	Expression
isfsuppf	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑊   𝑘,𝑍   𝜑,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝑉(𝑘)

Proof of Theorem isfsuppf

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfsuppf.fex	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		isfsuppf.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
3		isfsuppf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffund 5540	. 2 ⊢ (𝜑 → Fun 𝐹)
5		isfsuppf.w	. 2 ⊢ (𝜑 → 𝑊 ∈ Fin)
6		isfsuppf.wss	. . 3 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
7	3	fdmd 5543	. . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴)
8	6, 7	sseqtrrd 3287	. 2 ⊢ (𝜑 → 𝑊 ⊆ dom 𝐹)
9		fvelima 5757	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) → ∃𝑞 ∈ (dom 𝐹 ∖ 𝑊)(𝐹‘𝑞) = 𝑝)
10	4, 9	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) → ∃𝑞 ∈ (dom 𝐹 ∖ 𝑊)(𝐹‘𝑞) = 𝑝)
11		simprr 537	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → (𝐹‘𝑞) = 𝑝)
12		fveqeq2 5707	. . . . . . . 8 ⊢ (𝑘 = 𝑞 → ((𝐹‘𝑘) = 𝑍 ↔ (𝐹‘𝑞) = 𝑍))
13		isfupppf.k	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)
14	13	ralrimiva 2623	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑘) = 𝑍)
15	14	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → ∀𝑘 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑘) = 𝑍)
16		simprl 535	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → 𝑞 ∈ (dom 𝐹 ∖ 𝑊))
17	7	difeq1d 3346	. . . . . . . . . 10 ⊢ (𝜑 → (dom 𝐹 ∖ 𝑊) = (𝐴 ∖ 𝑊))
18	17	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → (dom 𝐹 ∖ 𝑊) = (𝐴 ∖ 𝑊))
19	16, 18	eleqtrd 2317	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → 𝑞 ∈ (𝐴 ∖ 𝑊))
20	12, 15, 19	rspcdva 2934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → (𝐹‘𝑞) = 𝑍)
21	11, 20	eqtr3d 2273	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → 𝑝 = 𝑍)
22		velsn 3726	. . . . . 6 ⊢ (𝑝 ∈ {𝑍} ↔ 𝑝 = 𝑍)
23	21, 22	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) ∧ (𝑞 ∈ (dom 𝐹 ∖ 𝑊) ∧ (𝐹‘𝑞) = 𝑝)) → 𝑝 ∈ {𝑍})
24	10, 23	rexlimddv 2673	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊))) → 𝑝 ∈ {𝑍})
25	24	ex 115	. . 3 ⊢ (𝜑 → (𝑝 ∈ (𝐹 “ (dom 𝐹 ∖ 𝑊)) → 𝑝 ∈ {𝑍}))
26	25	ssrdv 3254	. 2 ⊢ (𝜑 → (𝐹 “ (dom 𝐹 ∖ 𝑊)) ⊆ {𝑍})
27	1, 2, 4, 5, 8, 26	isfsuppfid 7297	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ∖ cdif 3217   ⊆ wss 3220  {csn 3709   class class class wbr 4131  dom cdm 4776   “ cima 4779  Fun wfun 5374  ⟶wf 5376  ‘cfv 5380  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-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  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-uni 3936  df-br 4132  df-opab 4194  df-id 4440  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-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-fsuppfi 7293

This theorem is used by:  psrbagfsuppd  15193  mplsubrgfi  15250