Theorem fsuppfiss 7300

Description: A finitely supported function has a finite subset of its domain such that values outside that subset map to zero. (Contributed by Jim Kingdon, 24-Oct-2026.)

Hypothesis

Ref	Expression
fsuppfund.1	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppfiss	⊢ (𝜑 → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))

Distinct variable groups:   𝑤,𝐹   𝑤,𝑍

Allowed substitution hint:   𝜑(𝑤)

Proof of Theorem fsuppfiss

Step	Hyp	Ref	Expression
1		fsuppfund.1	. . 3 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
2		relfsupp 7294	. . . . 5 ⊢ Rel finSupp
3	2	brrelex12i 4819	. . . 4 ⊢ (𝐹 finSupp 𝑍 → (𝐹 ∈ V ∧ 𝑍 ∈ V))
4		isfsuppfi 7296	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))))
5	1, 3, 4	3syl 17	. . 3 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))))
6	1, 5	mpbid 147	. 2 ⊢ (𝜑 → (Fun 𝐹 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})))
7	6	simprd 114	1 ⊢ (𝜑 → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   ∖ 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-ral 2533  df-rex 2534  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:  mplsubgfilemcl  15232  mplsubgfileminv  15233