Theorem funisfsuppdc 7298

Description: The property of a function to be finitely supported (in relation to a given zero). This shows that df-fsuppfi 7293 and df-supp 6480 give the same results, in the case of decidable equality. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 17-Oct-2026.)

Hypotheses

Ref	Expression
funisfsuppdc.r	|- ( ph -> F e. V )
funisfsuppdc.f	|- ( ph -> F : A --> B )
funisfsuppdc.z	|- ( ph -> Z e. B )
funisfsuppdc.dmdc	|- ( ph -> A. x e. A A. y e. A DECID x = y )
funisfsuppdc.cdmdc	|- ( ph -> A. x e. B A. y e. B DECID x = y )

Assertion

Ref	Expression
funisfsuppdc	|- ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )

Distinct variable groups:   x, A, y   x, B, y   x, F, y   y, Z

Allowed substitution hints:   ph( x, y)   V( x, y)   Z( x)

Proof of Theorem funisfsuppdc

Dummy variables r p w q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funisfsuppdc.r	. . . . 5 |- ( ph -> F e. V )
2		funisfsuppdc.z	. . . . 5 |- ( ph -> Z e. B )
3		isfsuppfi 7296	. . . . 5 |- ( ( F e. V /\ Z e. B ) -> ( F finSupp Z <-> ( Fun F /\ E. w ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) ) )
4	1, 2, 3	syl2anc 415	. . . 4 |- ( ph -> ( F finSupp Z <-> ( Fun F /\ E. w ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) ) )
5	4	simplbda 384	. . 3 |- ( ( ph /\ F finSupp Z ) -> E. w ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) )
6		simpr1 1034	. . . 4 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> w e. Fin )
7		funisfsuppdc.f	. . . . . 6 |- ( ph -> F : A --> B )
8	7	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> F : A --> B )
9		simplr3 1072	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( F " ( dom F \ w ) ) C_ { Z } )
10		simpr 110	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> q e. ( A \ w ) )
11	7	fdmd 5543	. . . . . . . . . . 11 |- ( ph -> dom F = A )
12	11	ad3antrrr 496	. . . . . . . . . 10 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> dom F = A )
13	12	difeq1d 3346	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( dom F \ w ) = ( A \ w ) )
14	10, 13	eleqtrrd 2318	. . . . . . . 8 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> q e. ( dom F \ w ) )
15	7	ffund 5540	. . . . . . . . . 10 |- ( ph -> Fun F )
16	15	ad3antrrr 496	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> Fun F )
17	10	eldifad 3231	. . . . . . . . . 10 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> q e. A )
18	17, 12	eleqtrrd 2318	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> q e. dom F )
19		funfvima 5953	. . . . . . . . 9 |- ( ( Fun F /\ q e. dom F ) -> ( q e. ( dom F \ w ) -> ( F ` q ) e. ( F " ( dom F \ w ) ) ) )
20	16, 18, 19	syl2anc 415	. . . . . . . 8 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( q e. ( dom F \ w ) -> ( F ` q ) e. ( F " ( dom F \ w ) ) ) )
21	14, 20	mpd 13	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( F ` q ) e. ( F " ( dom F \ w ) ) )
22	9, 21	sseldd 3249	. . . . . 6 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( F ` q ) e. { Z } )
23		elsni 3727	. . . . . 6 |- ( ( F ` q ) e. { Z } -> ( F ` q ) = Z )
24	22, 23	syl 14	. . . . 5 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ q e. ( A \ w ) ) -> ( F ` q ) = Z )
25		funisfsuppdc.dmdc	. . . . . . . 8 |- ( ph -> A. x e. A A. y e. A DECID x = y )
26	25	ad3antrrr 496	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ r e. A ) -> A. x e. A A. y e. A DECID x = y )
27		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ r e. A ) -> r e. A )
28		simpr2 1035	. . . . . . . . 9 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> w C_ dom F )
29	11	sseq2d 3278	. . . . . . . . . 10 |- ( ph -> ( w C_ dom F <-> w C_ A ) )
30	29	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> ( w C_ dom F <-> w C_ A ) )
31	28, 30	mpbid 147	. . . . . . . 8 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> w C_ A )
32	31	adantr 276	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ r e. A ) -> w C_ A )
33	6	adantr 276	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ r e. A ) -> w e. Fin )
34	26, 27, 32, 33	elssdc 7215	. . . . . 6 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ r e. A ) -> DECID r e. w )
35	34	ralrimiva 2623	. . . . 5 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> A. r e. A DECID r e. w )
36	8, 24, 35	suppssdc 6504	. . . 4 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> ( F supp Z ) C_ w )
37		eqeq2 2248	. . . . . . . . 9 |- ( y = Z -> ( ( F ` p ) = y <-> ( F ` p ) = Z ) )
38	37	dcbid 850	. . . . . . . 8 |- ( y = Z -> (DECID ( F ` p ) = y <-> DECID ( F ` p ) = Z ) )
39		eqeq1 2245	. . . . . . . . . . 11 |- ( x = ( F ` p ) -> ( x = y <-> ( F ` p ) = y ) )
40	39	dcbid 850	. . . . . . . . . 10 |- ( x = ( F ` p ) -> (DECID x = y <-> DECID ( F ` p ) = y ) )
41	40	ralbidv 2550	. . . . . . . . 9 |- ( x = ( F ` p ) -> ( A. y e. B DECID x = y <-> A. y e. B DECID ( F ` p ) = y ) )
42		funisfsuppdc.cdmdc	. . . . . . . . . 10 |- ( ph -> A. x e. B A. y e. B DECID x = y )
43	42	ad3antrrr 496	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> A. x e. B A. y e. B DECID x = y )
44	7	ad3antrrr 496	. . . . . . . . . 10 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> F : A --> B )
45	31	sselda 3248	. . . . . . . . . 10 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> p e. A )
46	44, 45	ffvelcdmd 5847	. . . . . . . . 9 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> ( F ` p ) e. B )
47	41, 43, 46	rspcdva 2934	. . . . . . . 8 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> A. y e. B DECID ( F ` p ) = y )
48	2	ad3antrrr 496	. . . . . . . 8 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> Z e. B )
49	38, 47, 48	rspcdva 2934	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> DECID ( F ` p ) = Z )
50	49	dcned 2426	. . . . . 6 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> DECID ( F ` p ) =/= Z )
51	7	ffnd 5537	. . . . . . . . . 10 |- ( ph -> F Fn A )
52		elsuppfng 6486	. . . . . . . . . 10 |- ( ( F Fn A /\ F e. V /\ Z e. B ) -> ( p e. ( F supp Z ) <-> ( p e. A /\ ( F ` p ) =/= Z ) ) )
53	51, 1, 2, 52	syl3anc 1278	. . . . . . . . 9 |- ( ph -> ( p e. ( F supp Z ) <-> ( p e. A /\ ( F ` p ) =/= Z ) ) )
54	53	ad3antrrr 496	. . . . . . . 8 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> ( p e. ( F supp Z ) <-> ( p e. A /\ ( F ` p ) =/= Z ) ) )
55	45, 54	mpbirand 445	. . . . . . 7 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> ( p e. ( F supp Z ) <-> ( F ` p ) =/= Z ) )
56	55	dcbid 850	. . . . . 6 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> (DECID p e. ( F supp Z ) <-> DECID ( F ` p ) =/= Z ) )
57	50, 56	mpbird 167	. . . . 5 |- ( ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) /\ p e. w ) -> DECID p e. ( F supp Z ) )
58	57	ralrimiva 2623	. . . 4 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> A. p e. w DECID p e. ( F supp Z ) )
59		ssfidc 7251	. . . 4 |- ( ( w e. Fin /\ ( F supp Z ) C_ w /\ A. p e. w DECID p e. ( F supp Z ) ) -> ( F supp Z ) e. Fin )
60	6, 36, 58, 59	syl3anc 1278	. . 3 |- ( ( ( ph /\ F finSupp Z ) /\ ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) ) -> ( F supp Z ) e. Fin )
61	5, 60	exlimddv 1954	. 2 |- ( ( ph /\ F finSupp Z ) -> ( F supp Z ) e. Fin )
62	1	adantr 276	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> F e. V )
63	2	adantr 276	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> Z e. B )
64	15	adantr 276	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> Fun F )
65		simpr 110	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> ( F supp Z ) e. Fin )
66		suppssdmg 6493	. . . 4 |- ( ( F e. V /\ Z e. B ) -> ( F supp Z ) C_ dom F )
67	62, 63, 66	syl2anc 415	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> ( F supp Z ) C_ dom F )
68	11	difeq1d 3346	. . . . . 6 |- ( ph -> ( dom F \ ( F supp Z ) ) = ( A \ ( F supp Z ) ) )
69	68	imaeq2d 5129	. . . . 5 |- ( ph -> ( F " ( dom F \ ( F supp Z ) ) ) = ( F " ( A \ ( F supp Z ) ) ) )
70	7, 1, 2, 42	suppinisegdc 6513	. . . . . . . 8 |- ( ph -> ( A \ ( F supp Z ) ) = ( `' F " { Z } ) )
71	70	imaeq2d 5129	. . . . . . 7 |- ( ph -> ( F " ( A \ ( F supp Z ) ) ) = ( F " ( `' F " { Z } ) ) )
72		funimacnv 5460	. . . . . . . 8 |- ( Fun F -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
73	15, 72	syl 14	. . . . . . 7 |- ( ph -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
74	71, 73	eqtrd 2271	. . . . . 6 |- ( ph -> ( F " ( A \ ( F supp Z ) ) ) = ( { Z } i^i ran F ) )
75		inss1 3451	. . . . . 6 |- ( { Z } i^i ran F ) C_ { Z }
76	74, 75	eqsstrdi 3300	. . . . 5 |- ( ph -> ( F " ( A \ ( F supp Z ) ) ) C_ { Z } )
77	69, 76	eqsstrd 3284	. . . 4 |- ( ph -> ( F " ( dom F \ ( F supp Z ) ) ) C_ { Z } )
78	77	adantr 276	. . 3 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> ( F " ( dom F \ ( F supp Z ) ) ) C_ { Z } )
79	62, 63, 64, 65, 67, 78	isfsuppfid 7297	. 2 |- ( ( ph /\ ( F supp Z ) e. Fin ) -> F finSupp Z )
80	61, 79	impbida 604	1 |- ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   A.wral 2528   \ cdif 3217   i^i cin 3219   C_ wss 3220   {csn 3709   class class class wbr 4131   `'ccnv 4775   dom cdm 4776   ran crn 4777   "cima 4779   Fun wfun 5374   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   supp csupp 6479   Fincfn 7028   finSupp cfsupp 7292

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  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-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480  df-1o 6692  df-er 6812  df-en 7029  df-fin 7031  df-fsuppfi 7293

This theorem is used by:  fcdmnn0fsuppfi  9742