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Theorem fnsuppeq0 6946
Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
Assertion
Ref Expression
fnsuppeq0  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( ( F supp  Z
)  =  (/)  <->  F  =  ( A  X.  { Z } ) ) )

Proof of Theorem fnsuppeq0
StepHypRef Expression
1 ss0b 3789 . . 3  |-  ( ( F supp  Z )  C_  (/)  <->  ( F supp  Z )  =  (/) )
2 un0 3784 . . . . . . . 8  |-  ( A  u.  (/) )  =  A
3 uncom 3607 . . . . . . . 8  |-  ( A  u.  (/) )  =  (
(/)  u.  A )
42, 3eqtr3i 2451 . . . . . . 7  |-  A  =  ( (/)  u.  A
)
54fneq2i 5681 . . . . . 6  |-  ( F  Fn  A  <->  F  Fn  ( (/)  u.  A ) )
65biimpi 197 . . . . 5  |-  ( F  Fn  A  ->  F  Fn  ( (/)  u.  A
) )
763ad2ant1 1026 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  F  Fn  ( (/)  u.  A ) )
8 fnex 6139 . . . . 5  |-  ( ( F  Fn  A  /\  A  e.  W )  ->  F  e.  _V )
983adant3 1025 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  F  e.  _V )
10 simp3 1007 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  Z  e.  V )
11 incom 3652 . . . . . 6  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
12 in0 3785 . . . . . 6  |-  ( A  i^i  (/) )  =  (/)
1311, 12eqtri 2449 . . . . 5  |-  ( (/)  i^i 
A )  =  (/)
1413a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( (/)  i^i  A )  =  (/) )
15 fnsuppres 6945 . . . 4  |-  ( ( F  Fn  ( (/)  u.  A )  /\  ( F  e.  _V  /\  Z  e.  V )  /\  ( (/) 
i^i  A )  =  (/) )  ->  ( ( F supp  Z )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
167, 9, 10, 14, 15syl121anc 1269 . . 3  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( ( F supp  Z
)  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
171, 16syl5bbr 262 . 2  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( ( F supp  Z
)  =  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
18 fnresdm 5695 . . . 4  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
19183ad2ant1 1026 . . 3  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( F  |`  A )  =  F )
2019eqeq1d 2422 . 2  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( ( F  |`  A )  =  ( A  X.  { Z } )  <->  F  =  ( A  X.  { Z } ) ) )
2117, 20bitrd 256 1  |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V )  ->  ( ( F supp  Z
)  =  (/)  <->  F  =  ( A  X.  { Z } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    i^i cin 3432    C_ wss 3433   (/)c0 3758   {csn 3993    X. cxp 4844    |` cres 4848    Fn wfn 5588  (class class class)co 6297   supp csupp 6917
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-supp 6918
This theorem is referenced by:  fczsupp0  6947  cantnf0  8177  mdegldg  22992  mdeg0  22996
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