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Theorem fnsuppeq0 5956
Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Assertion
Ref Expression
fnsuppeq0  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  =  (/) 
<->  F  =  ( A  X.  { Z }
) ) )

Proof of Theorem fnsuppeq0
StepHypRef Expression
1 ss0b 3659 . . 3  |-  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) )
2 un0 3654 . . . . . . . 8  |-  ( A  u.  (/) )  =  A
3 uncom 3493 . . . . . . . 8  |-  ( A  u.  (/) )  =  (
(/)  u.  A )
42, 3eqtr3i 2460 . . . . . . 7  |-  A  =  ( (/)  u.  A
)
54fneq2i 5543 . . . . . 6  |-  ( F  Fn  A  <->  F  Fn  ( (/)  u.  A ) )
65biimpi 188 . . . . 5  |-  ( F  Fn  A  ->  F  Fn  ( (/)  u.  A
) )
76adantr 453 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  F  Fn  ( (/)  u.  A ) )
8 incom 3535 . . . . . 6  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
9 in0 3655 . . . . . 6  |-  ( A  i^i  (/) )  =  (/)
108, 9eqtri 2458 . . . . 5  |-  ( (/)  i^i 
A )  =  (/)
1110a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( (/)  i^i  A )  =  (/) )
12 simpr 449 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  Z  e.  V )
13 fnsuppres 5955 . . . 4  |-  ( ( F  Fn  ( (/)  u.  A )  /\  ( (/) 
i^i  A )  =  (/)  /\  Z  e.  V
)  ->  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
147, 11, 12, 13syl3anc 1185 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
151, 14syl5bbr 252 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  =  (/) 
<->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
16 fnresdm 5557 . . . 4  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1716adantr 453 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( F  |`  A )  =  F )
1817eqeq1d 2446 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( F  |`  A )  =  ( A  X.  { Z } )  <->  F  =  ( A  X.  { Z } ) ) )
1915, 18bitrd 246 1  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  =  (/) 
<->  F  =  ( A  X.  { Z }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816    X. cxp 4879   `'ccnv 4880    |` cres 4883   "cima 4884    Fn wfn 5452
This theorem is referenced by:  mdegldg  19994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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