Theorem ressuppfi 9464

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 2746	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺)
3	2	oveq1d 7463	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4		ressuppfi.s	. . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
5	3, 4	eqeltrd 2844	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin)
6		ressuppfi.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
7		unfi 9238	. . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin)
8	5, 6, 7	syl2anc 583	. 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 8226	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
12	9, 10, 11	syl2anc 583	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
13	8, 12	ssfid 9329	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  dom cdm 5700   ↾ cres 5702  (class class class)co 7448   supp csupp 8201  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-supp 8202  df-1o 8522  df-en 9004  df-fin 9007

This theorem is referenced by:  resfsupp  9465