Theorem fsuppcorn 7291

Description: The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the ( F supp Z ) C_ ran G condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7236. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)

Hypotheses

Ref	Expression
fsuppco.f	|- ( ph -> F finSupp Z )
fsuppco.g	|- ( ph -> G : X -1-1-> Y )
fsuppco.z	|- ( ph -> Z e. W )
fsuppco.v	|- ( ph -> F e. V )
fsuppcorn.g	|- ( ph -> G e. U )
fsuppcorn.rn	|- ( ph -> ( F supp Z ) C_ ran G )

Assertion

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

Proof of Theorem fsuppcorn

Step	Hyp	Ref	Expression
1		fsuppco.v	. . 3 |- ( ph -> F e. V )
2		fsuppco.g	. . . 4 |- ( ph -> G : X -1-1-> Y )
3		df-f1 5377	. . . . 5 |- ( G : X -1-1-> Y <-> ( G : X --> Y /\ Fun `' G ) )
4	3	simprbi 275	. . . 4 |- ( G : X -1-1-> Y -> Fun `' G )
5	2, 4	syl 14	. . 3 |- ( ph -> Fun `' G )
6		cofunex2g 6329	. . 3 |- ( ( F e. V /\ Fun `' G ) -> ( F o. G ) e. _V )
7	1, 5, 6	syl2anc 415	. 2 |- ( ph -> ( F o. G ) e. _V )
8		fsuppco.z	. 2 |- ( ph -> Z e. W )
9		fsuppco.f	. . . 4 |- ( ph -> F finSupp Z )
10	9	fsuppfund 7284	. . 3 |- ( ph -> Fun F )
11		f1fun 5596	. . . 4 |- ( G : X -1-1-> Y -> Fun G )
12	2, 11	syl 14	. . 3 |- ( ph -> Fun G )
13		funco 5412	. . 3 |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
14	10, 12, 13	syl2anc 415	. 2 |- ( ph -> Fun ( F o. G ) )
15		fsuppcorn.g	. . . 4 |- ( ph -> G e. U )
16		suppcofn 6496	. . . 4 |- ( ( ( F e. V /\ G e. U ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
17	1, 15, 10, 12, 16	syl22anc 1279	. . 3 |- ( ph -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
18	9	fsuppimpd 7283	. . . 4 |- ( ph -> ( F supp Z ) e. Fin )
19		f1cnv 5658	. . . . . . 7 |- ( G : X -1-1-> Y -> `' G : ran G -1-1-onto-> X )
20	2, 19	syl 14	. . . . . 6 |- ( ph -> `' G : ran G -1-1-onto-> X )
21		f1of1 5633	. . . . . 6 |- ( `' G : ran G -1-1-onto-> X -> `' G : ran G -1-1-> X )
22	20, 21	syl 14	. . . . 5 |- ( ph -> `' G : ran G -1-1-> X )
23		fsuppcorn.rn	. . . . 5 |- ( ph -> ( F supp Z ) C_ ran G )
24		f1imaeng 7069	. . . . 5 |- ( ( `' G : ran G -1-1-> X /\ ( F supp Z ) C_ ran G /\ ( F supp Z ) e. Fin ) -> ( `' G " ( F supp Z ) ) ~~ ( F supp Z ) )
25	22, 23, 18, 24	syl3anc 1278	. . . 4 |- ( ph -> ( `' G " ( F supp Z ) ) ~~ ( F supp Z ) )
26		enfii 7166	. . . 4 |- ( ( ( F supp Z ) e. Fin /\ ( `' G " ( F supp Z ) ) ~~ ( F supp Z ) ) -> ( `' G " ( F supp Z ) ) e. Fin )
27	18, 25, 26	syl2anc 415	. . 3 |- ( ph -> ( `' G " ( F supp Z ) ) e. Fin )
28	17, 27	eqeltrd 2315	. 2 |- ( ph -> ( ( F o. G ) supp Z ) e. Fin )
29	7, 8, 14, 28	isfsuppd 7280	1 |- ( ph -> ( F o. G ) finSupp Z )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   `'ccnv 4768   ran crn 4770   "cima 4772   o. ccom 4773   Fun wfun 5366   -->wf 5368   -1-1->wf1 5369   -1-1-onto->wf1o 5371  (class class class)co 6075   supp csupp 6465   ~~ cen 7010   Fincfn 7012   finSupp cfsupp 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-supp 6466  df-er 6797  df-en 7013  df-fin 7015  df-fsupp 7276

This theorem is referenced by: (None)