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Theorem fnsuppeq0 5920
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 3625 . . 3  |-  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) )
2 un0 3620 . . . . . . . 8  |-  ( A  u.  (/) )  =  A
3 uncom 3459 . . . . . . . 8  |-  ( A  u.  (/) )  =  (
(/)  u.  A )
42, 3eqtr3i 2434 . . . . . . 7  |-  A  =  ( (/)  u.  A
)
54fneq2i 5507 . . . . . 6  |-  ( F  Fn  A  <->  F  Fn  ( (/)  u.  A ) )
65biimpi 187 . . . . 5  |-  ( F  Fn  A  ->  F  Fn  ( (/)  u.  A
) )
76adantr 452 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  F  Fn  ( (/)  u.  A ) )
8 incom 3501 . . . . . 6  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
9 in0 3621 . . . . . 6  |-  ( A  i^i  (/) )  =  (/)
108, 9eqtri 2432 . . . . 5  |-  ( (/)  i^i 
A )  =  (/)
1110a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( (/)  i^i  A )  =  (/) )
12 simpr 448 . . . 4  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  Z  e.  V )
13 fnsuppres 5919 . . . 4  |-  ( ( F  Fn  ( (/)  u.  A )  /\  ( (/) 
i^i  A )  =  (/)  /\  Z  e.  V
)  ->  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
147, 11, 12, 13syl3anc 1184 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
151, 14syl5bbr 251 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } ) )  =  (/) 
<->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
16 fnresdm 5521 . . . 4  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1716adantr 452 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( F  |`  A )  =  F )
1817eqeq1d 2420 . 2  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( F  |`  A )  =  ( A  X.  { Z } )  <->  F  =  ( A  X.  { Z } ) ) )
1915, 18bitrd 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    \ cdif 3285    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782    X. cxp 4843   `'ccnv 4844    |` cres 4847   "cima 4848    Fn wfn 5416
This theorem is referenced by:  mdegldg  19950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429
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