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Theorem fsuppimp 7789
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 7785 . . . 4  |-  Rel finSupp
21brrelexi 4983 . . 3  |-  ( R finSupp  Z  ->  R  e.  _V )
31brrelex2i 4984 . . 3  |-  ( R finSupp  Z  ->  Z  e.  _V )
42, 3jca 530 . 2  |-  ( R finSupp  Z  ->  ( R  e. 
_V  /\  Z  e.  _V ) )
5 isfsupp 7787 . . 3  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
65biimpd 207 . 2  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) ) )
74, 6mpcom 34 1  |-  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   _Vcvv 3058   class class class wbr 4394   Fun wfun 5519  (class class class)co 6234   supp csupp 6856   Fincfn 7474   finSupp cfsupp 7783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-iota 5489  df-fun 5527  df-fv 5533  df-ov 6237  df-fsupp 7784
This theorem is referenced by:  fsuppimpd  7790  fsuppunfi  7803  fsuppunbi  7804  fsuppres  7808  fsuppco  7815  oemapvali  8055  mptnn0fsuppr  12059  gsumzres  17130  gsumzf1o  17133
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