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Theorem sniffsupp 7881
Description: A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)
Hypotheses
Ref Expression
sniffsupp.i  |-  ( ph  ->  I  e.  V )
sniffsupp.0  |-  ( ph  ->  .0.  e.  W )
sniffsupp.f  |-  F  =  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )
Assertion
Ref Expression
sniffsupp  |-  ( ph  ->  F finSupp  .0.  )
Distinct variable groups:    x, I    x, X    x,  .0.    ph, x
Allowed substitution hints:    A( x)    F( x)    V( x)    W( x)

Proof of Theorem sniffsupp
StepHypRef Expression
1 sniffsupp.f . 2  |-  F  =  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )
2 snfi 7608 . . . 4  |-  { X }  e.  Fin
3 eldifsni 4159 . . . . . . . 8  |-  ( x  e.  ( I  \  { X } )  ->  x  =/=  X )
43adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  { X } ) )  ->  x  =/=  X )
54neneqd 2669 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  { X } ) )  ->  -.  x  =  X
)
6 iffalse 3954 . . . . . 6  |-  ( -.  x  =  X  ->  if ( x  =  X ,  A ,  .0.  )  =  .0.  )
75, 6syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  { X } ) )  ->  if ( x  =  X ,  A ,  .0.  )  =  .0.  )
8 sniffsupp.i . . . . 5  |-  ( ph  ->  I  e.  V )
97, 8suppss2 6946 . . . 4  |-  ( ph  ->  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) supp  .0.  )  C_  { X }
)
10 ssfi 7752 . . . 4  |-  ( ( { X }  e.  Fin  /\  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  )
) supp  .0.  )  C_  { X } )  -> 
( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) supp  .0.  )  e.  Fin )
112, 9, 10sylancr 663 . . 3  |-  ( ph  ->  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) supp  .0.  )  e.  Fin )
12 funmpt 5630 . . . . 5  |-  Fun  (
x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )
1312a1i 11 . . . 4  |-  ( ph  ->  Fun  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) )
14 mptexg 6141 . . . . 5  |-  ( I  e.  V  ->  (
x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )  e.  _V )
158, 14syl 16 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )  e.  _V )
16 sniffsupp.0 . . . 4  |-  ( ph  ->  .0.  e.  W )
17 funisfsupp 7846 . . . 4  |-  ( ( Fun  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )  /\  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) )  e.  _V  /\  .0.  e.  W )  ->  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  )
) finSupp  .0.  <->  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  )
) supp  .0.  )  e.  Fin ) )
1813, 15, 16, 17syl3anc 1228 . . 3  |-  ( ph  ->  ( ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) finSupp  .0.  <->  (
( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) supp  .0.  )  e.  Fin ) )
1911, 18mpbird 232 . 2  |-  ( ph  ->  ( x  e.  I  |->  if ( x  =  X ,  A ,  .0.  ) ) finSupp  .0.  )
201, 19syl5eqbr 4486 1  |-  ( ph  ->  F finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    \ cdif 3478    C_ wss 3481   ifcif 3945   {csn 4033   class class class wbr 4453    |-> cmpt 4511   Fun wfun 5588  (class class class)co 6295   supp csupp 6913   Fincfn 7528   finSupp cfsupp 7841
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-supp 6914  df-1o 7142  df-er 7323  df-en 7529  df-fin 7532  df-fsupp 7842
This theorem is referenced by:  dprdfid  16929  snifpsrbag  17885
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