Theorem resfsupp 8302

Description: If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)

Hypotheses

Ref	Expression
resfsupp.b	|- ( ph -> ( dom F \ B ) e. Fin )
resfsupp.e	|- ( ph -> F e. W )
resfsupp.f	|- ( ph -> Fun F )
resfsupp.g	|- ( ph -> G = ( F |` B ) )
resfsupp.s	|- ( ph -> G finSupp Z )
resfsupp.z	|- ( ph -> Z e. V )

Assertion

Ref	Expression
resfsupp	|- ( ph -> F finSupp Z )

Proof of Theorem resfsupp

Step	Hyp	Ref	Expression
1		resfsupp.b	. . 3 |- ( ph -> ( dom F \ B ) e. Fin )
2		resfsupp.e	. . 3 |- ( ph -> F e. W )
3		resfsupp.g	. . 3 |- ( ph -> G = ( F |` B ) )
4		resfsupp.s	. . . 4 |- ( ph -> G finSupp Z )
5	4	fsuppimpd 8282	. . 3 |- ( ph -> ( G supp Z ) e. Fin )
6		resfsupp.z	. . 3 |- ( ph -> Z e. V )
7	1, 2, 3, 5, 6	ressuppfi 8301	. 2 |- ( ph -> ( F supp Z ) e. Fin )
8		resfsupp.f	. . 3 |- ( ph -> Fun F )
9		funisfsupp 8280	. . 3 |- ( ( Fun F /\ F e. W /\ Z e. V ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
10	8, 2, 6, 9	syl3anc 1326	. 2 |- ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
11	7, 10	mpbird 247	1 |- ( ph -> F finSupp Z )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   \ cdif 3571   class class class wbr 4653   dom cdm 5114   |` 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-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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  lincext2  42244