Theorem fsuppco2 8308

Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8309 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppco2.z	|- ( ph -> Z e. W )
fsuppco2.f	|- ( ph -> F : A --> B )
fsuppco2.g	|- ( ph -> G : B --> B )
fsuppco2.a	|- ( ph -> A e. U )
fsuppco2.b	|- ( ph -> B e. V )
fsuppco2.n	|- ( ph -> F finSupp Z )
fsuppco2.i	|- ( ph -> ( G ` Z ) = Z )

Assertion

Ref	Expression
fsuppco2	|- ( ph -> ( G o. F ) finSupp Z )

Proof of Theorem fsuppco2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppco2.g	. . . 4 |- ( ph -> G : B --> B )
2		ffun 6048	. . . 4 |- ( G : B --> B -> Fun G )
3	1, 2	syl 17	. . 3 |- ( ph -> Fun G )
4		fsuppco2.f	. . . 4 |- ( ph -> F : A --> B )
5		ffun 6048	. . . 4 |- ( F : A --> B -> Fun F )
6	4, 5	syl 17	. . 3 |- ( ph -> Fun F )
7		funco 5928	. . 3 |- ( ( Fun G /\ Fun F ) -> Fun ( G o. F ) )
8	3, 6, 7	syl2anc 693	. 2 |- ( ph -> Fun ( G o. F ) )
9		fsuppco2.n	. . . 4 |- ( ph -> F finSupp Z )
10	9	fsuppimpd 8282	. . 3 |- ( ph -> ( F supp Z ) e. Fin )
11		fco 6058	. . . . 5 |- ( ( G : B --> B /\ F : A --> B ) -> ( G o. F ) : A --> B )
12	1, 4, 11	syl2anc 693	. . . 4 |- ( ph -> ( G o. F ) : A --> B )
13		eldifi 3732	. . . . . 6 |- ( x e. ( A \ ( F supp Z ) ) -> x e. A )
14		fvco3 6275	. . . . . 6 |- ( ( F : A --> B /\ x e. A ) -> ( ( G o. F ) ` x ) = ( G ` ( F ` x ) ) )
15	4, 13, 14	syl2an 494	. . . . 5 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( ( G o. F ) ` x ) = ( G ` ( F ` x ) ) )
16		ssid 3624	. . . . . . . 8 |- ( F supp Z ) C_ ( F supp Z )
17	16	a1i 11	. . . . . . 7 |- ( ph -> ( F supp Z ) C_ ( F supp Z ) )
18		fsuppco2.a	. . . . . . 7 |- ( ph -> A e. U )
19		fsuppco2.z	. . . . . . 7 |- ( ph -> Z e. W )
20	4, 17, 18, 19	suppssr 7326	. . . . . 6 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( F ` x ) = Z )
21	20	fveq2d 6195	. . . . 5 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( G ` ( F ` x ) ) = ( G ` Z ) )
22		fsuppco2.i	. . . . . 6 |- ( ph -> ( G ` Z ) = Z )
23	22	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( G ` Z ) = Z )
24	15, 21, 23	3eqtrd 2660	. . . 4 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( ( G o. F ) ` x ) = Z )
25	12, 24	suppss 7325	. . 3 |- ( ph -> ( ( G o. F ) supp Z ) C_ ( F supp Z ) )
26		ssfi 8180	. . 3 |- ( ( ( F supp Z ) e. Fin /\ ( ( G o. F ) supp Z ) C_ ( F supp Z ) ) -> ( ( G o. F ) supp Z ) e. Fin )
27	10, 25, 26	syl2anc 693	. 2 |- ( ph -> ( ( G o. F ) supp Z ) e. Fin )
28		fsuppco2.b	. . . . 5 |- ( ph -> B e. V )
29		fex 6490	. . . . 5 |- ( ( G : B --> B /\ B e. V ) -> G e. _V )
30	1, 28, 29	syl2anc 693	. . . 4 |- ( ph -> G e. _V )
31		fex 6490	. . . . 5 |- ( ( F : A --> B /\ A e. U ) -> F e. _V )
32	4, 18, 31	syl2anc 693	. . . 4 |- ( ph -> F e. _V )
33		coexg 7117	. . . 4 |- ( ( G e. _V /\ F e. _V ) -> ( G o. F ) e. _V )
34	30, 32, 33	syl2anc 693	. . 3 |- ( ph -> ( G o. F ) e. _V )
35		isfsupp 8279	. . 3 |- ( ( ( G o. F ) e. _V /\ Z e. W ) -> ( ( G o. F ) finSupp Z <-> ( Fun ( G o. F ) /\ ( ( G o. F ) supp Z ) e. Fin ) ) )
36	34, 19, 35	syl2anc 693	. 2 |- ( ph -> ( ( G o. F ) finSupp Z <-> ( Fun ( G o. F ) /\ ( ( G o. F ) supp Z ) e. Fin ) ) )
37	8, 27, 36	mpbir2and 957	1 |- ( ph -> ( G o. F ) finSupp Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (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-rep 4771  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-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-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:  gsumzinv  18345  gsumsub  18348