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Theorem fsuppimp 7826
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
Dummy variables  z 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fsupp 7821 . . . . 5  |- finSupp  =  { <. r ,  z >.  |  ( Fun  r  /\  ( r supp  z )  e.  Fin ) }
21relopabi 5121 . . . 4  |-  Rel finSupp
32brrelexi 5034 . . 3  |-  ( R finSupp  Z  ->  R  e.  _V )
42brrelex2i 5035 . . 3  |-  ( R finSupp  Z  ->  Z  e.  _V )
53, 4jca 532 . 2  |-  ( R finSupp  Z  ->  ( R  e. 
_V  /\  Z  e.  _V ) )
6 isfsupp 7824 . . 3  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  <->  ( Fun  R  /\  ( R supp  Z
)  e.  Fin )
) )
76biimpd 207 . 2  |-  ( ( R  e.  _V  /\  Z  e.  _V )  ->  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) ) )
85, 7mpcom 36 1  |-  ( R finSupp  Z  ->  ( Fun  R  /\  ( R supp  Z )  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   _Vcvv 3108   class class class wbr 4442   Fun wfun 5575  (class class class)co 6277   supp csupp 6893   Fincfn 7508   finSupp cfsupp 7820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-fsupp 7821
This theorem is referenced by:  fsuppimpd  7827  fsuppunfi  7840  fsuppunbi  7841  fsuppres  7845  resfsupp  7847  fsuppco  7852  oemapvali  8094  mptnn0fsuppr  12063  gsumzres  16700  gsumzf1o  16703  gsumzadd  16721  mdegldg  22196
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