Theorem fsuppres 8300

Description: The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
fsuppres.s	|- ( ph -> F finSupp Z )
fsuppres.z	|- ( ph -> Z e. V )

Assertion

Ref	Expression
fsuppres	|- ( ph -> ( F |` X ) finSupp Z )

Proof of Theorem fsuppres

Step	Hyp	Ref	Expression
1		fsuppres.s	. . 3 |- ( ph -> F finSupp Z )
2		fsuppimp 8281	. . . 4 |- ( F finSupp Z -> ( Fun F /\ ( F supp Z ) e. Fin ) )
3		relprcnfsupp 8278	. . . . . . . . . . . 12 |- ( -. F e. _V -> -. F finSupp Z )
4	3	con4i 113	. . . . . . . . . . 11 |- ( F finSupp Z -> F e. _V )
5	1, 4	syl 17	. . . . . . . . . 10 |- ( ph -> F e. _V )
6		fsuppres.z	. . . . . . . . . 10 |- ( ph -> Z e. V )
7	5, 6	jca 554	. . . . . . . . 9 |- ( ph -> ( F e. _V /\ Z e. V ) )
8	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( F e. _V /\ Z e. V ) )
9		ressuppss 7314	. . . . . . . 8 |- ( ( F e. _V /\ Z e. V ) -> ( ( F |` X ) supp Z ) C_ ( F supp Z ) )
10		ssfi 8180	. . . . . . . . 9 |- ( ( ( F supp Z ) e. Fin /\ ( ( F |` X ) supp Z ) C_ ( F supp Z ) ) -> ( ( F |` X ) supp Z ) e. Fin )
11	10	expcom 451	. . . . . . . 8 |- ( ( ( F |` X ) supp Z ) C_ ( F supp Z ) -> ( ( F supp Z ) e. Fin -> ( ( F |` X ) supp Z ) e. Fin ) )
12	8, 9, 11	3syl 18	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( ( F supp Z ) e. Fin -> ( ( F |` X ) supp Z ) e. Fin ) )
13	12	expcom 451	. . . . . 6 |- ( Fun F -> ( ph -> ( ( F supp Z ) e. Fin -> ( ( F |` X ) supp Z ) e. Fin ) ) )
14	13	com23 86	. . . . 5 |- ( Fun F -> ( ( F supp Z ) e. Fin -> ( ph -> ( ( F |` X ) supp Z ) e. Fin ) ) )
15	14	imp 445	. . . 4 |- ( ( Fun F /\ ( F supp Z ) e. Fin ) -> ( ph -> ( ( F |` X ) supp Z ) e. Fin ) )
16	2, 15	syl 17	. . 3 |- ( F finSupp Z -> ( ph -> ( ( F |` X ) supp Z ) e. Fin ) )
17	1, 16	mpcom 38	. 2 |- ( ph -> ( ( F |` X ) supp Z ) e. Fin )
18		funres 5929	. . . . 5 |- ( Fun F -> Fun ( F |` X ) )
19	18	adantr 481	. . . 4 |- ( ( Fun F /\ ( F supp Z ) e. Fin ) -> Fun ( F |` X ) )
20	1, 2, 19	3syl 18	. . 3 |- ( ph -> Fun ( F |` X ) )
21		resexg 5442	. . . 4 |- ( F e. _V -> ( F |` X ) e. _V )
22	1, 4, 21	3syl 18	. . 3 |- ( ph -> ( F |` X ) e. _V )
23		funisfsupp 8280	. . 3 |- ( ( Fun ( F |` X ) /\ ( F |` X ) e. _V /\ Z e. V ) -> ( ( F |` X ) finSupp Z <-> ( ( F |` X ) supp Z ) e. Fin ) )
24	20, 22, 6, 23	syl3anc 1326	. 2 |- ( ph -> ( ( F |` X ) finSupp Z <-> ( ( F |` X ) supp Z ) e. Fin ) )
25	17, 24	mpbird 247	1 |- ( ph -> ( F |` X ) finSupp Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |` cres 5116   Fun wfun 5882  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275

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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-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-supp 7296  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  dprdfadd  18419  frlmsplit2  20112  gsumle  29779  lindslinindimp2lem3  42249  lindslinindsimp2lem5  42251