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Theorem fnsuppeq0 5733
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 3484 . . 3  |-  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) )
2 un0 3479 . . . . . . . 8  |-  ( A  u.  (/) )  =  A
3 uncom 3319 . . . . . . . 8  |-  ( A  u.  (/) )  =  (
(/)  u.  A )
42, 3eqtr3i 2305 . . . . . . 7  |-  A  =  ( (/)  u.  A
)
54fneq2i 5339 . . . . . 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 3361 . . . . . 6  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
9 in0 3480 . . . . . 6  |-  ( A  i^i  (/) )  =  (/)
108, 9eqtri 2303 . . . . 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 5732 . . . 4  |-  ( ( F  Fn  ( (/)  u.  A )  /\  ( (/) 
i^i  A )  =  (/)  /\  Z  e.  V
)  ->  ( ( `' F " ( _V 
\  { Z }
) )  C_  (/)  <->  ( F  |`  A )  =  ( A  X.  { Z } ) ) )
147, 11, 12, 13syl3anc 1182 . . 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 5353 . . . 4  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1716adantr 451 . . 3  |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( F  |`  A )  =  F )
1817eqeq1d 2291 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250
This theorem is referenced by:  mdegldg  19452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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