Theorem isfsupp 7214

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 5353	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 276	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		oveq12 6037	. . . 4 |- ( ( r = R /\ z = Z ) -> ( r supp z ) = ( R supp Z ) )
4	3	eleq1d 2300	. . 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 7211	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
7	5, 6	brabga 4364	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 2202   class class class wbr 4093   Fun wfun 5327  (class class class)co 6028   supp csupp 6413   Fincfn 6952   finSupp cfsupp 7210

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-fsupp 7211

This theorem is referenced by:  isfsuppd  7215  funisfsupp  7216  fsuppimp  7217