Theorem isfsuppfi 7296

Description: The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)

Assertion

Ref	Expression
isfsuppfi	|- ( ( 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 } ) ) ) )

Distinct variable groups:   w, R   w, Z

Allowed substitution hints:   U( w)   V( w)

Proof of Theorem isfsuppfi

Dummy variables r z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funeq 5400	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 276	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		dmeq 4983	. . . . . . 7 |- ( r = R -> dom r = dom R )
4	3	adantr 276	. . . . . 6 |- ( ( r = R /\ z = Z ) -> dom r = dom R )
5	4	sseq2d 3278	. . . . 5 |- ( ( r = R /\ z = Z ) -> ( w C_ dom r <-> w C_ dom R ) )
6		simpl 109	. . . . . . 7 |- ( ( r = R /\ z = Z ) -> r = R )
7	4	difeq1d 3346	. . . . . . 7 |- ( ( r = R /\ z = Z ) -> ( dom r \ w ) = ( dom R \ w ) )
8	6, 7	imaeq12d 5130	. . . . . 6 |- ( ( r = R /\ z = Z ) -> ( r " ( dom r \ w ) ) = ( R " ( dom R \ w ) ) )
9		sneq 3720	. . . . . . 7 |- ( z = Z -> { z } = { Z } )
10	9	adantl 277	. . . . . 6 |- ( ( r = R /\ z = Z ) -> { z } = { Z } )
11	8, 10	sseq12d 3279	. . . . 5 |- ( ( r = R /\ z = Z ) -> ( ( r " ( dom r \ w ) ) C_ { z } <-> ( R " ( dom R \ w ) ) C_ { Z } ) )
12	5, 11	3anbi23d 1356	. . . 4 |- ( ( r = R /\ z = Z ) -> ( ( 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 } ) ) )
13	12	exbidv 1878	. . 3 |- ( ( r = R /\ z = Z ) -> ( E. w ( 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 } ) ) )
14	2, 13	anbi12d 477	. 2 |- ( ( r = R /\ z = Z ) -> ( ( Fun r /\ E. w ( w e. Fin /\ w C_ dom r /\ ( r " ( dom r \ w ) ) C_ { z } ) ) <-> ( Fun R /\ E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) ) )
15		df-fsuppfi 7293	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ E. w ( w e. Fin /\ w C_ dom r /\ ( r " ( dom r \ w ) ) C_ { z } ) ) }
16	14, 15	brabga 4408	1 |- ( ( 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 } ) ) ) )

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-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-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:  isfsuppfid  7297  funisfsuppdc  7298  fsuppfund  7299