Theorem isfsuppf 7298

Description: Showing that a function is finitely supported. (Contributed by Jim Kingdon, 2-Nov-2026.)

Hypotheses

Ref	Expression
isfsuppf.f	|- ( ph -> F : A --> B )
isfsuppf.fex	|- ( ph -> F e. V )
isfsuppf.z	|- ( ph -> Z e. B )
isfsuppf.w	|- ( ph -> W e. Fin )
isfsuppf.wss	|- ( ph -> W C_ A )
isfupppf.k	|- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )

Assertion

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

Distinct variable groups:   A, k   k, F   k, W   k, Z   ph, k

Allowed substitution hints:   B( k)   V( k)

Proof of Theorem isfsuppf

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

Step	Hyp	Ref	Expression
1		isfsuppf.fex	. 2 |- ( ph -> F e. V )
2		isfsuppf.z	. 2 |- ( ph -> Z e. B )
3		isfsuppf.f	. . 3 |- ( ph -> F : A --> B )
4	3	ffund 5540	. 2 |- ( ph -> Fun F )
5		isfsuppf.w	. 2 |- ( ph -> W e. Fin )
6		isfsuppf.wss	. . 3 |- ( ph -> W C_ A )
7	3	fdmd 5543	. . 3 |- ( ph -> dom F = A )
8	6, 7	sseqtrrd 3287	. 2 |- ( ph -> W C_ dom F )
9		fvelima 5757	. . . . . 6 |- ( ( Fun F /\ p e. ( F " ( dom F \ W ) ) ) -> E. q e. ( dom F \ W ) ( F ` q ) = p )
10	4, 9	sylan 283	. . . . 5 |- ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) -> E. q e. ( dom F \ W ) ( F ` q ) = p )
11		simprr 537	. . . . . . 7 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> ( F ` q ) = p )
12		fveqeq2 5707	. . . . . . . 8 |- ( k = q -> ( ( F ` k ) = Z <-> ( F ` q ) = Z ) )
13		isfupppf.k	. . . . . . . . . 10 |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
14	13	ralrimiva 2623	. . . . . . . . 9 |- ( ph -> A. k e. ( A \ W ) ( F ` k ) = Z )
15	14	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> A. k e. ( A \ W ) ( F ` k ) = Z )
16		simprl 535	. . . . . . . . 9 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> q e. ( dom F \ W ) )
17	7	difeq1d 3346	. . . . . . . . . 10 |- ( ph -> ( dom F \ W ) = ( A \ W ) )
18	17	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> ( dom F \ W ) = ( A \ W ) )
19	16, 18	eleqtrd 2317	. . . . . . . 8 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> q e. ( A \ W ) )
20	12, 15, 19	rspcdva 2934	. . . . . . 7 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> ( F ` q ) = Z )
21	11, 20	eqtr3d 2273	. . . . . 6 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> p = Z )
22		velsn 3726	. . . . . 6 |- ( p e. { Z } <-> p = Z )
23	21, 22	sylibr 134	. . . . 5 |- ( ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) /\ ( q e. ( dom F \ W ) /\ ( F ` q ) = p ) ) -> p e. { Z } )
24	10, 23	rexlimddv 2673	. . . 4 |- ( ( ph /\ p e. ( F " ( dom F \ W ) ) ) -> p e. { Z } )
25	24	ex 115	. . 3 |- ( ph -> ( p e. ( F " ( dom F \ W ) ) -> p e. { Z } ) )
26	25	ssrdv 3254	. 2 |- ( ph -> ( F " ( dom F \ W ) ) C_ { Z } )
27	1, 2, 4, 5, 8, 26	isfsuppfid 7297	1 |- ( ph -> F finSupp Z )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   \ cdif 3217   C_ wss 3220   {csn 3709   class class class wbr 4131   dom cdm 4776   "cima 4779   Fun wfun 5374   -->wf 5376   ` cfv 5380   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-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  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-fv 5388  df-fsuppfi 7293

This theorem is used by:  psrbagfsuppd  15193  mplsubrgfi  15250