Theorem isfsuppfid 7297

Description: Deduction form of isfsuppfi 7296. (Contributed by SN, 29-Jul-2024.) (Revised by Jim Kingdon, 16-Oct-2026.)

Hypotheses

Ref	Expression
isfsuppfid.r	|- ( ph -> R e. V )
isfsuppfid.z	|- ( ph -> Z e. U )
isfsuppfid.1	|- ( ph -> Fun R )
isfsuppfid.w	|- ( ph -> W e. Fin )
isfsuppfid.wss	|- ( ph -> W C_ dom R )
isfsuppfid.wz	|- ( ph -> ( R " ( dom R \ W ) ) C_ { Z } )

Assertion

Ref	Expression
isfsuppfid	|- ( ph -> R finSupp Z )

Proof of Theorem isfsuppfid

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfsuppfid.1	. 2 |- ( ph -> Fun R )
2		isfsuppfid.w	. . 3 |- ( ph -> W e. Fin )
3		isfsuppfid.wss	. . 3 |- ( ph -> W C_ dom R )
4		isfsuppfid.wz	. . 3 |- ( ph -> ( R " ( dom R \ W ) ) C_ { Z } )
5		eleq1 2301	. . . . . 6 |- ( w = W -> ( w e. Fin <-> W e. Fin ) )
6		sseq1 3271	. . . . . 6 |- ( w = W -> ( w C_ dom R <-> W C_ dom R ) )
7		difeq2 3341	. . . . . . . 8 |- ( w = W -> ( dom R \ w ) = ( dom R \ W ) )
8	7	imaeq2d 5129	. . . . . . 7 |- ( w = W -> ( R " ( dom R \ w ) ) = ( R " ( dom R \ W ) ) )
9	8	sseq1d 3277	. . . . . 6 |- ( w = W -> ( ( R " ( dom R \ w ) ) C_ { Z } <-> ( R " ( dom R \ W ) ) C_ { Z } ) )
10	5, 6, 9	3anbi123d 1353	. . . . 5 |- ( w = W -> ( ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) <-> ( W e. Fin /\ W C_ dom R /\ ( R " ( dom R \ W ) ) C_ { Z } ) ) )
11	10	spcegv 2913	. . . 4 |- ( W e. Fin -> ( ( W e. Fin /\ W C_ dom R /\ ( R " ( dom R \ W ) ) C_ { Z } ) -> E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) )
12	2, 11	syl 14	. . 3 |- ( ph -> ( ( W e. Fin /\ W C_ dom R /\ ( R " ( dom R \ W ) ) C_ { Z } ) -> E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) )
13	2, 3, 4, 12	mp3and 1381	. 2 |- ( ph -> E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) )
14		isfsuppfid.r	. . 3 |- ( ph -> R e. V )
15		isfsuppfid.z	. . 3 |- ( ph -> Z e. U )
16		isfsuppfi 7296	. . 3 |- ( ( R e. V /\ Z e. U ) -> ( R finSupp Z <-> ( Fun R /\ E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) ) )
17	14, 15, 16	syl2anc 415	. 2 |- ( ph -> ( R finSupp Z <-> ( Fun R /\ E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) ) )
18	1, 13, 17	mpbir2and 957	1 |- ( ph -> R finSupp Z )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   \ cdif 3217   C_ wss 3220   {csn 3709   class class class wbr 4131   dom cdm 4776   "cima 4779   Fun wfun 5374   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-rab 2537  df-v 2823  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-br 4132  df-opab 4194  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-fun 5382  df-fsuppfi 7293

This theorem is used by:  funisfsuppdc  7298  fdmfifsupp  7300  fczfsuppd  7303  0fsupp  7304  snopfsupp  7305