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Theorem fvn0elsupp 6803
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.)
Assertion
Ref Expression
fvn0elsupp  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )

Proof of Theorem fvn0elsupp
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( B  e.  V  /\  X  e.  B )  ->  X  e.  B )
2 simpr 461 . . 3  |-  ( ( G : B --> A  /\  ( G `  X )  =/=  (/) )  ->  ( G `  X )  =/=  (/) )
31, 2anim12i 566 . 2  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  -> 
( X  e.  B  /\  ( G `  X
)  =/=  (/) ) )
4 ffn 5654 . . . 4  |-  ( G : B --> A  ->  G  Fn  B )
54ad2antrl 727 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  G  Fn  B )
6 simpll 753 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  B  e.  V )
7 0ex 4517 . . . 4  |-  (/)  e.  _V
87a1i 11 . . 3  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  (/) 
e.  _V )
9 elsuppfn 6795 . . 3  |-  ( ( G  Fn  B  /\  B  e.  V  /\  (/) 
e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `
 X )  =/=  (/) ) ) )
105, 6, 8, 9syl3anc 1219 . 2  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  -> 
( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
113, 10mpbird 232 1  |-  ( ( ( B  e.  V  /\  X  e.  B
)  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2642   _Vcvv 3065   (/)c0 3732    Fn wfn 5508   -->wf 5509   ` cfv 5513  (class class class)co 6187   supp csupp 6787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-supp 6788
This theorem is referenced by:  oemapvali  7990  cantnflem1c  7993
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