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Theorem resfnfinfin 40255
Description: The restriction of a function by 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 5788 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 479 . . 3 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → Rel 𝐹)
3 resisresindm 40224 . . 3 (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
42, 3syl 17 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
5 fnfun 5787 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
6 funfn 5718 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
75, 6sylib 206 . . . 4 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
8 fnresin2 5805 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9 infi 7943 . . . . . 6 (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10 fnfi 7997 . . . . . 6 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
119, 10sylan2 489 . . . . 5 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
1211ex 448 . . . 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 443 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
154, 14eqeltrd 2592 1 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  cin 3443  dom cdm 4932  cres 4934  Rel wrel 4937  Fun wfun 5683   Fn wfn 5684  Fincfn 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-oadd 7325  df-er 7503  df-en 7716  df-fin 7719
This theorem is referenced by:  residfi  40256
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