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Theorem resfnfinfin 31736
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 5672 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
21adantr 465 . . 3  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  Rel  F )
3 resisresindm 31731 . . 3  |-  ( Rel 
F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
42, 3syl 16 . 2  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F
) ) )
5 fnfun 5671 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
6 funfn 5610 . . . . 5  |-  ( Fun 
F  <->  F  Fn  dom  F )
75, 6sylib 196 . . . 4  |-  ( F  Fn  A  ->  F  Fn  dom  F )
8 fnresin2 5689 . . . 4  |-  ( F  Fn  dom  F  -> 
( F  |`  ( B  i^i  dom  F )
)  Fn  ( B  i^i  dom  F )
)
9 infi 7735 . . . . . 6  |-  ( B  e.  Fin  ->  ( B  i^i  dom  F )  e.  Fin )
10 fnfi 7789 . . . . . 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 20 . . 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 2550 1  |-  ( ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3470   dom cdm 4994    |` cres 4996   Rel wrel 4999   Fun wfun 5575    Fn wfn 5576   Fincfn 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-fin 7512
This theorem is referenced by:  residfi  31740
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