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Theorem fsuppimp 7898
Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
Assertion
Ref Expression
fsuppimp  |-  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) )

Proof of Theorem fsuppimp
StepHypRef Expression
1 relfsupp 7894 . . . 4  |-  Rel finSupp
21brrelexi 4894 . . 3  |-  ( R finSupp  Z  ->  R  e.  _V )
31brrelex2i 4895 . . 3  |-  ( R finSupp  Z  ->  Z  e.  _V )
42, 3jca 534 . 2  |-  ( R finSupp  Z  ->  ( R  e. 
_V  /\  Z  e.  _V ) )
5 isfsupp 7896 . . 3  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
65biimpd 210 . 2  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) ) )
74, 6mpcom 37 1  |-  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1872   _Vcvv 3080   class class class wbr 4423   Fun wfun 5595  (class class class)co 6305   supp csupp 6925   Fincfn 7580   finSupp cfsupp 7892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-fsupp 7893
This theorem is referenced by:  fsuppimpd  7899  fsuppunfi  7912  fsuppunbi  7913  fsuppres  7917  fsuppco  7924  oemapvali  8197  mptnn0fsuppr  12217  gsumzres  17542  gsumzf1o  17545
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