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Theorem fvn0elsupp 6917
Description: If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
Assertion
Ref Expression
fvn0elsupp  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )

Proof of Theorem fvn0elsupp
StepHypRef Expression
1 simpr 459 . . 3  |-  ( ( B  e.  V  /\  X  e.  B )  ->  X  e.  B )
2 simpr 459 . . 3  |-  ( ( G  Fn  B  /\  ( G `  X )  =/=  (/) )  ->  ( G `  X )  =/=  (/) )
31, 2anim12i 564 . 2  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  -> 
( X  e.  B  /\  ( G `  X
)  =/=  (/) ) )
4 simprl 756 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  G  Fn  B )
5 simpll 752 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  B  e.  V )
6 0ex 4525 . . . 4  |-  (/)  e.  _V
76a1i 11 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  (/) 
e.  _V )
8 elsuppfn 6909 . . 3  |-  ( ( G  Fn  B  /\  B  e.  V  /\  (/) 
e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `
 X )  =/=  (/) ) ) )
94, 5, 7, 8syl3anc 1230 . 2  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  -> 
( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
103, 9mpbird 232 1  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1842    =/= wne 2598   _Vcvv 3058   (/)c0 3737    Fn wfn 5563   ` cfv 5568  (class class class)co 6277   supp csupp 6901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-supp 6902
This theorem is referenced by:  fvn0elsuppb  6919  oemapvali  8134  cantnflem1c  8137
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