Theorem resfnfinfin 9249

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 6602	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → Rel 𝐹)
3		resindm 5997	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵))
4	3	eqcomd 2743	. . 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 6600	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
7	6	funfnd 6531	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
8		fnresin2 6626	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9		infi 9182	. . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10		fnfi 9114	. . . . . 6 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
11	9, 10	sylan2 594	. . . . 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 2837	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3902  dom cdm 5632   ↾ cres 5634  Rel wrel 5637   Fn wfn 6495  Fincfn 8895

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 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-en 8896  df-fin 8899

This theorem is referenced by:  residfi  9250  itg1addlem4  25668  gsumhashmul  33161  pthhashvtx  35344  imadomfi  42372