Theorem isfsupp 7255

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 5377	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 276	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		oveq12 6067	. . . 4 |- ( ( r = R /\ z = Z ) -> ( r supp z ) = ( R supp Z ) )
4	3	eleq1d 2303	. . 3 |- ( ( r = R /\ z = Z ) -> ( ( r supp z ) e. Fin <-> ( R supp Z ) e. Fin ) )
5	2, 4	anbi12d 473	. 2 |- ( ( r = R /\ z = Z ) -> ( ( Fun r /\ ( r supp z ) e. Fin ) <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
6		df-fsupp 7252	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
7	5, 6	brabga 4387	1 |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   class class class wbr 4114   Fun wfun 5351  (class class class)co 6058   supp csupp 6448   Fincfn 6988   finSupp cfsupp 7251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-fsupp 7252

This theorem is referenced by:  isfsuppd  7256  funisfsupp  7257  fsuppimp  7258