Theorem resfsupp 9155

Description: If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)

Hypotheses

Ref	Expression
resfsupp.b	⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
resfsupp.e	⊢ (𝜑 → 𝐹 ∈ 𝑊)
resfsupp.f	⊢ (𝜑 → Fun 𝐹)
resfsupp.g	⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))
resfsupp.s	⊢ (𝜑 → 𝐺 finSupp 𝑍)
resfsupp.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)

Assertion

Ref	Expression
resfsupp	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Proof of Theorem resfsupp

Step	Hyp	Ref	Expression
1		resfsupp.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
2		resfsupp.e	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
3		resfsupp.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))
4		resfsupp.s	. . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍)
5	4	fsuppimpd 9135	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
6		resfsupp.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
7	1, 2, 3, 5, 6	ressuppfi 9154	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
8		resfsupp.f	. . 3 ⊢ (𝜑 → Fun 𝐹)
9		funisfsupp 9133	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
10	8, 2, 6, 9	syl3anc 1370	. 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))
11	7, 10	mpbird 256	1 ⊢ (𝜑 → 𝐹 finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  Fun wfun 6427  (class class class)co 7275   supp csupp 7977  Fincfn 8733   finSupp cfsupp 9128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-supp 7978  df-1o 8297  df-en 8734  df-fin 8737  df-fsupp 9129

This theorem is referenced by:  lincext2  45796