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Theorem resfnfinfin 9351
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 6651 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 479 . . 3 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → Rel 𝐹)
3 resindm 6030 . . . 4 (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹𝐵))
43eqcomd 2731 . . 3 (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
52, 4syl 17 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
6 fnfun 6649 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
76funfnd 6579 . . . 4 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
8 fnresin2 6676 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9 infi 9286 . . . . . 6 (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10 fnfi 9199 . . . . . 6 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
119, 10sylan2 591 . . . . 5 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
1211ex 411 . . . 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 405 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
155, 14eqeltrd 2825 1 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cin 3940  dom cdm 5673  cres 5675  Rel wrel 5678   Fn wfn 6538  Fincfn 8957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7866  df-1o 8480  df-en 8958  df-fin 8961
This theorem is referenced by:  residfi  9352  itg1addlem4  25641  gsumhashmul  32810  pthhashvtx  34790  imadomfi  41525
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