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Theorem resfnfinfin 37956
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  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )

Proof of Theorem resfnfinfin
StepHypRef Expression
1 fnrel 5662 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
21adantr 465 . . 3  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  Rel  F )
3 resisresindm 37951 . . 3  |-  ( Rel 
F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
42, 3syl 17 . 2  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F
) ) )
5 fnfun 5661 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
6 funfn 5600 . . . . 5  |-  ( Fun 
F  <->  F  Fn  dom  F )
75, 6sylib 198 . . . 4  |-  ( F  Fn  A  ->  F  Fn  dom  F )
8 fnresin2 5679 . . . 4  |-  ( F  Fn  dom  F  -> 
( F  |`  ( B  i^i  dom  F )
)  Fn  ( B  i^i  dom  F )
)
9 infi 7780 . . . . . 6  |-  ( B  e.  Fin  ->  ( B  i^i  dom  F )  e.  Fin )
10 fnfi 7834 . . . . . 6  |-  ( ( ( F  |`  ( B  i^i  dom  F )
)  Fn  ( B  i^i  dom  F )  /\  ( B  i^i  dom  F )  e.  Fin )  ->  ( F  |`  ( B  i^i  dom  F )
)  e.  Fin )
119, 10sylan2 474 . . . . 5  |-  ( ( ( F  |`  ( B  i^i  dom  F )
)  Fn  ( B  i^i  dom  F )  /\  B  e.  Fin )  ->  ( F  |`  ( B  i^i  dom  F
) )  e.  Fin )
1211ex 434 . . . 4  |-  ( ( F  |`  ( B  i^i  dom  F ) )  Fn  ( B  i^i  dom 
F )  ->  ( B  e.  Fin  ->  ( F  |`  ( B  i^i  dom 
F ) )  e. 
Fin ) )
137, 8, 123syl 18 . . 3  |-  ( F  Fn  A  ->  ( B  e.  Fin  ->  ( F  |`  ( B  i^i  dom 
F ) )  e. 
Fin ) )
1413imp 429 . 2  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  ( B  i^i  dom  F )
)  e.  Fin )
154, 14eqeltrd 2492 1  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    i^i cin 3415   dom cdm 4825    |` cres 4827   Rel wrel 4830   Fun wfun 5565    Fn wfn 5566   Fincfn 7556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-fin 7560
This theorem is referenced by:  residfi  37960
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