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Theorem fsuppsssupp 7841
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 753 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  G  e.  V
)
2 simplr 754 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  Fun  G )
3 relfsupp 7827 . . . 4  |-  Rel finSupp
43brrelex2i 5040 . . 3  |-  ( F finSupp  Z  ->  Z  e.  _V )
54ad2antrl 727 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  Z  e.  _V )
6 id 22 . . . . 5  |-  ( F finSupp  Z  ->  F finSupp  Z )
76fsuppimpd 7832 . . . 4  |-  ( F finSupp  Z  ->  ( F supp  Z
)  e.  Fin )
87anim1i 568 . . 3  |-  ( ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) )  -> 
( ( F supp  Z
)  e.  Fin  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )
98adantl 466 . 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 7840 . 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 1231 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 369    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   Fun wfun 5580  (class class class)co 6282   supp csupp 6898   Fincfn 7513   finSupp cfsupp 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-om 6679  df-er 7308  df-en 7514  df-fin 7517  df-fsupp 7826
This theorem is referenced by:  cantnflem1d  8103  cantnflem1  8104  dprdfinv  16849  dmdprdsplitlem  16874  dpjidcl  16897  frlmphllem  18578  frlmphl  18579  rrxcph  21559  tdeglem4  22193
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