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Theorem fsuppsssupp 7914
Description: If the support of a function is a subset of the support of a finitely supported function, the function is finitely supported. (Contributed by AV, 2-Jul-2019.) (Proof shortened by AV, 15-Jul-2019.)
Assertion
Ref Expression
fsuppsssupp  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  G finSupp  Z )

Proof of Theorem fsuppsssupp
StepHypRef Expression
1 simpll 759 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  G  e.  V
)
2 simplr 761 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  Fun  G )
3 relfsupp 7900 . . . 4  |-  Rel finSupp
43brrelex2i 4901 . . 3  |-  ( F finSupp  Z  ->  Z  e.  _V )
54ad2antrl 733 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  Z  e.  _V )
6 id 23 . . . . 5  |-  ( F finSupp  Z  ->  F finSupp  Z )
76fsuppimpd 7905 . . . 4  |-  ( F finSupp  Z  ->  ( F supp  Z
)  e.  Fin )
87anim1i 571 . . 3  |-  ( ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) )  -> 
( ( F supp  Z
)  e.  Fin  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )
98adantl 468 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  ( ( F supp 
Z )  e.  Fin  /\  ( G supp  Z ) 
C_  ( F supp  Z
) ) )
10 suppssfifsupp 7913 . 2  |-  ( ( ( G  e.  V  /\  Fun  G  /\  Z  e.  _V )  /\  (
( F supp  Z )  e.  Fin  /\  ( G supp 
Z )  C_  ( F supp  Z ) ) )  ->  G finSupp  Z )
111, 2, 5, 9, 10syl31anc 1268 1  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  G finSupp  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1873   _Vcvv 3085    C_ wss 3442   class class class wbr 4429   Fun wfun 5601  (class class class)co 6311   supp csupp 6931   Fincfn 7586   finSupp cfsupp 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3087  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-br 4430  df-opab 4489  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-ov 6314  df-om 6713  df-er 7380  df-en 7587  df-fin 7590  df-fsupp 7899
This theorem is referenced by:  cantnflem1  8208  dprdfinv  17657  dmdprdsplitlem  17675  dpjidcl  17696  frlmphllem  19342  frlmphl  19343  rrxcph  22355  tdeglem4  23013
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