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Theorem fnsuppeq0 5853
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 3572 . . 3  |-  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) )
2 un0 3567 . . . . . . . 8  |-  ( A  u.  (/) )  =  A
3 uncom 3407 . . . . . . . 8  |-  ( A  u.  (/) )  =  (
(/)  u.  A )
42, 3eqtr3i 2388 . . . . . . 7  |-  A  =  ( (/)  u.  A
)
54fneq2i 5444 . . . . . 6  |-  ( F  Fn  A  <->  F  Fn  ( (/)  u.  A ) )
65biimpi 186 . . . . 5  |-  ( F  Fn  A  ->  F  Fn  ( (/)  u.  A
) )
76adantr 451 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  F  Fn  ( (/)  u.  A ) )
8 incom 3449 . . . . . 6  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
9 in0 3568 . . . . . 6  |-  ( A  i^i  (/) )  =  (/)
108, 9eqtri 2386 . . . . 5  |-  ( (/)  i^i 
A )  =  (/)
1110a1i 10 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( (/)  i^i  A )  =  (/) )
12 simpr 447 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  Z  e.  V )
13 fnsuppres 5852 . . . 4  |-  ( ( F  Fn  ( (/)  u.  A )  /\  ( (/) 
i^i  A )  =  (/)  /\  Z  e.  V
)  ->  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
147, 11, 12, 13syl3anc 1183 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
151, 14syl5bbr 250 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  =  (/) 
<->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
16 fnresdm 5458 . . . 4  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1716adantr 451 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( F  |`  A )  =  F )
1817eqeq1d 2374 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( F  |`  A )  =  ( A  X.  { Z } )  <->  F  =  ( A  X.  { Z } ) ) )
1915, 18bitrd 244 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 176    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    \ cdif 3235    u. cun 3236    i^i cin 3237    C_ wss 3238   (/)c0 3543   {csn 3729    X. cxp 4790   `'ccnv 4791    |` cres 4794   "cima 4795    Fn wfn 5353
This theorem is referenced by:  mdegldg  19667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366
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