Theorem ressuppfi 9354

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 2767	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺)
3	2	oveq1d 7425	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍))
4		ressuppfi.s	. . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)
5	3, 4	eqeltrd 2861	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin)
6		ressuppfi.b	. . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)
7		unfi 9154	. . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin)
8	5, 6, 7	syl2anc 595	. 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 8180	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
12	9, 10, 11	syl2anc 595	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
13	8, 12	ssfid 9228	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  dom cdm 5661   ↾ cres 5663  (class class class)co 7410   supp csupp 8155  Fincfn 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-supp 8156  df-1o 8452  df-en 8943  df-fin 8946

This theorem is referenced by:  resfsupp  9355