Theorem isfsupp 7242

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 5372	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 276	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		oveq12 6059	. . . 4 |- ( ( r = R /\ z = Z ) -> ( r supp z ) = ( R supp Z ) )
4	3	eleq1d 2301	. . 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 7239	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
7	5, 6	brabga 4382	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 2203   class class class wbr 4109   Fun wfun 5346  (class class class)co 6050   supp csupp 6435   Fincfn 6975   finSupp cfsupp 7238

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-fsupp 7239

This theorem is referenced by:  isfsuppd  7243  funisfsupp  7244  fsuppimp  7245