Theorem resfnfinfin 9328

Description: The restriction of a function to a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)

Assertion

Ref	Expression
resfnfinfin	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)

Proof of Theorem resfnfinfin

Step	Hyp	Ref	Expression
1		fnrel 6648	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	adantr 481	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → Rel 𝐹)
3		resindm 6028	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵))
4	3	eqcomd 2738	. . 3 ⊢ (Rel 𝐹 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
5	2, 4	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
6		fnfun 6646	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
7	6	funfnd 6576	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
8		fnresin2 6673	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9		infi 9264	. . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10		fnfi 9177	. . . . . 6 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
11	9, 10	sylan2 593	. . . . 5 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
12	11	ex 413	. . . 4 ⊢ ((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin))
13	7, 8, 12	3syl 18	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin))
14	13	imp 407	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
15	5, 14	eqeltrd 2833	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3946  dom cdm 5675   ↾ cres 5677  Rel wrel 5680   Fn wfn 6535  Fincfn 8935

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-fin 8939

This theorem is referenced by:  residfi  9329  itg1addlem4  25207  gsumhashmul  32195  pthhashvtx  34106