Theorem resfnfinfin 8246

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

Step	Hyp	Ref	Expression
1		fnrel 5989	. . . 4 |- ( F Fn A -> Rel F )
2	1	adantr 481	. . 3 |- ( ( F Fn A /\ B e. Fin ) -> Rel F )
3		resindm 5444	. . . 4 |- ( Rel F -> ( F |` ( B i^i dom F ) ) = ( F |` B ) )
4	3	eqcomd 2628	. . 3 |- ( Rel F -> ( F |` B ) = ( F |` ( B i^i dom F ) ) )
5	2, 4	syl 17	. 2 |- ( ( F Fn A /\ B e. Fin ) -> ( F |` B ) = ( F |` ( B i^i dom F ) ) )
6		fnfun 5988	. . . . 5 |- ( F Fn A -> Fun F )
7		funfn 5918	. . . . 5 |- ( Fun F <-> F Fn dom F )
8	6, 7	sylib 208	. . . 4 |- ( F Fn A -> F Fn dom F )
9		fnresin2 6006	. . . 4 |- ( F Fn dom F -> ( F |` ( B i^i dom F ) ) Fn ( B i^i dom F ) )
10		infi 8184	. . . . . 6 |- ( B e. Fin -> ( B i^i dom F ) e. Fin )
11		fnfi 8238	. . . . . 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 )
12	10, 11	sylan2 491	. . . . 5 |- ( ( ( F |` ( B i^i dom F ) ) Fn ( B i^i dom F ) /\ B e. Fin ) -> ( F |` ( B i^i dom F ) ) e. Fin )
13	12	ex 450	. . . 4 |- ( ( F |` ( B i^i dom F ) ) Fn ( B i^i dom F ) -> ( B e. Fin -> ( F |` ( B i^i dom F ) ) e. Fin ) )
14	8, 9, 13	3syl 18	. . 3 |- ( F Fn A -> ( B e. Fin -> ( F |` ( B i^i dom F ) ) e. Fin ) )
15	14	imp 445	. 2 |- ( ( F Fn A /\ B e. Fin ) -> ( F |` ( B i^i dom F ) ) e. Fin )
16	5, 15	eqeltrd 2701	1 |- ( ( F Fn A /\ B e. Fin ) -> ( F |` B ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   dom cdm 5114   |` cres 5116   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959

This theorem is referenced by:  residfi  8247