Theorem resfnfinfin 9237

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 6594	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → Rel 𝐹)
3		resindm 5989	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵))
4	3	eqcomd 2742	. . 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 6592	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
7	6	funfnd 6523	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
8		fnresin2 6618	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9		infi 9170	. . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10		fnfi 9102	. . . . . 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 412	. . . 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 406	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
15	5, 14	eqeltrd 2836	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∩ cin 3900  dom cdm 5624   ↾ cres 5626  Rel wrel 5629   Fn wfn 6487  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-fin 8887

This theorem is referenced by:  residfi  9238  itg1addlem4  25656  gsumhashmul  33150  pthhashvtx  35322  imadomfi  42256