Theorem isfsupp 7289

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
isfsupp	|- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )

Proof of Theorem isfsupp

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

Step	Hyp	Ref	Expression
1		funeq 5397	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 276	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		oveq12 6094	. . . 4 |- ( ( r = R /\ z = Z ) -> ( r supp z ) = ( R supp Z ) )
4	3	eleq1d 2307	. . 3 |- ( ( r = R /\ z = Z ) -> ( ( r supp z ) e. Fin <-> ( R supp Z ) e. Fin ) )
5	2, 4	anbi12d 477	. 2 |- ( ( r = R /\ z = Z ) -> ( ( Fun r /\ ( r supp z ) e. Fin ) <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
6		df-fsupp 7286	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
7	5, 6	brabga 4406	1 |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   Fun wfun 5371  (class class class)co 6085   supp csupp 6475   Fincfn 7022   finSupp cfsupp 7285

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 4249  ax-pow 4311  ax-pr 4346

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-rex 2534  df-v 2823  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 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-fsupp 7286

This theorem is used by:  isfsuppd  7290  funisfsupp  7291  fsuppimp  7292