Theorem isfsupp 8279

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 5908	. . . 4 |- ( r = R -> ( Fun r <-> Fun R ) )
2	1	adantr 481	. . 3 |- ( ( r = R /\ z = Z ) -> ( Fun r <-> Fun R ) )
3		oveq12 6659	. . . 4 |- ( ( r = R /\ z = Z ) -> ( r supp z ) = ( R supp Z ) )
4	3	eleq1d 2686	. . 3 |- ( ( r = R /\ z = Z ) -> ( ( r supp z ) e. Fin <-> ( R supp Z ) e. Fin ) )
5	2, 4	anbi12d 747	. 2 |- ( ( r = R /\ z = Z ) -> ( ( Fun r /\ ( r supp z ) e. Fin ) <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
6		df-fsupp 8276	. 2 |- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
7	5, 6	brabga 4989	1 |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   Fun wfun 5882  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-fsupp 8276

This theorem is referenced by:  funisfsupp  8280  fsuppimp  8281  fdmfifsupp  8285  fczfsuppd  8293  fsuppmptif  8305  fsuppco2  8308  fsuppcor  8309  gsumzadd  18322  gsumpt  18361  gsum2dlem2  18370  gsum2d  18371  gsum2d2lem  18372  rmfsupp  42155  mndpfsupp  42157  scmfsupp  42159  mptcfsupp  42161