MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsuppsssupp Structured version   Unicode version

Theorem fsuppsssupp 7905
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 758 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  G  e.  V
)
2 simplr 760 . 2  |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )  ->  Fun  G )
3 relfsupp 7891 . . . 4  |-  Rel finSupp
43brrelex2i 4896 . . 3  |-  ( F finSupp  Z  ->  Z  e.  _V )
54ad2antrl 732 . 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 7896 . . . 4  |-  ( F finSupp  Z  ->  ( F supp  Z
)  e.  Fin )
87anim1i 570 . . 3  |-  ( ( F finSupp  Z  /\  ( G supp  Z )  C_  ( F supp  Z ) )  -> 
( ( F supp  Z
)  e.  Fin  /\  ( G supp  Z )  C_  ( F supp  Z ) ) )
98adantl 467 . 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 7904 . 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 1267 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 370    e. wcel 1870   _Vcvv 3087    C_ wss 3442   class class class wbr 4426   Fun wfun 5595  (class class class)co 6305   supp csupp 6925   Fincfn 7577   finSupp cfsupp 7889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-om 6707  df-er 7371  df-en 7578  df-fin 7581  df-fsupp 7890
This theorem is referenced by:  cantnflem1  8193  dprdfinv  17587  dmdprdsplitlem  17605  dpjidcl  17626  frlmphllem  19269  frlmphl  19270  rrxcph  22244  tdeglem4  22886
  Copyright terms: Public domain W3C validator