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Theorem fsuppsssupp 7746
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 7732 . . . 4  |-  Rel finSupp
43brrelex2i 4987 . . 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 7737 . . . 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 7745 . 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 1222 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 1758   _Vcvv 3076    C_ wss 3435   class class class wbr 4399   Fun wfun 5519  (class class class)co 6199   supp csupp 6799   Fincfn 7419   finSupp cfsupp 7730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-om 6586  df-er 7210  df-en 7420  df-fin 7423  df-fsupp 7731
This theorem is referenced by:  cantnflem1d  8006  cantnflem1  8007  dprdfinv  16630  dmdprdsplitlem  16655  dpjidcl  16678  frlmphllem  18329  frlmphl  18330  rrxcph  21027  tdeglem4  21661
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