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Theorem resfnfinfin 9346
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 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)

Proof of Theorem resfnfinfin
StepHypRef Expression
1 fnrel 6650 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 480 . . 3 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → Rel 𝐹)
3 resindm 6028 . . . 4 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
43eqcomd 2733 . . 3 (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
52, 4syl 17 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
6 fnfun 6648 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
76funfnd 6578 . . . 4 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
8 fnresin2 6675 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9 infi 9282 . . . . . 6 (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10 fnfi 9195 . . . . . 6 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
119, 10sylan2 592 . . . . 5 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
1211ex 412 . . . 4 ((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin))
137, 8, 123syl 18 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin))
1413imp 406 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
155, 14eqeltrd 2828 1 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cin 3943  dom cdm 5672  cres 5674  Rel wrel 5677   Fn wfn 6537  Fincfn 8953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7863  df-1o 8478  df-en 8954  df-fin 8957
This theorem is referenced by:  residfi  9347  itg1addlem4  25602  gsumhashmul  32735  pthhashvtx  34660  imadomfi  41397
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