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Theorem resfnfinfin 40339
 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 5903 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 480 . . 3 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → Rel 𝐹)
3 resisresindm 40320 . . 3 (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
42, 3syl 17 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
5 fnfun 5902 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
6 funfn 5833 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
75, 6sylib 207 . . . 4 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
8 fnresin2 5920 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹))
9 infi 8069 . . . . . 6 (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin)
10 fnfi 8123 . . . . . 6 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
119, 10sylan2 490 . . . . 5 (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
1211ex 449 . . . 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 444 . 2 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)
154, 14eqeltrd 2688 1 ((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845 This theorem is referenced by:  residfi  40340
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