Theorem ressuppfi 9300

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 2741	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺)
3	2	oveq1d 7373	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4		ressuppfi.s	. . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
5	3, 4	eqeltrd 2835	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin)
6		ressuppfi.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
7		unfi 9097	. . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin)
8	5, 6, 7	syl2anc 585	. 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 8127	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
12	9, 10, 11	syl2anc 585	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
13	8, 12	ssfid 9171	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3897   ∪ cun 3898   ⊆ wss 3900  dom cdm 5623   ↾ cres 5625  (class class class)co 7358   supp csupp 8102  Fincfn 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-supp 8103  df-1o 8397  df-en 8886  df-fin 8889

This theorem is referenced by:  resfsupp  9301