Theorem ressuppfi 9011

Description: If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.)

Hypotheses

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

Assertion

Ref	Expression
ressuppfi	⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Proof of Theorem ressuppfi

Step	Hyp	Ref	Expression
1		ressuppfi.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))
2	1	eqcomd 2743	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺)
3	2	oveq1d 7228	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4		ressuppfi.s	. . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
5	3, 4	eqeltrd 2838	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin)
6		ressuppfi.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
7		unfi 8850	. . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin)
8	5, 6, 7	syl2anc 587	. 2 ⊢ (𝜑 → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin)
9		ressuppfi.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
10		ressuppfi.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
11		ressuppssdif 7927	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
12	9, 10, 11	syl2anc 587	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
13	8, 12	ssfid 8898	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863   ∪ cun 3864   ⊆ wss 3866  dom cdm 5551   ↾ cres 5553  (class class class)co 7213   supp csupp 7903  Fincfn 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-supp 7904  df-1o 8202  df-en 8627  df-fin 8630

This theorem is referenced by:  resfsupp  9012