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Theorem elsuppfn 6803
Description: An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
Assertion
Ref Expression
elsuppfn  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( S  e.  ( F supp  Z )  <->  ( S  e.  X  /\  ( F `  S )  =/=  Z ) ) )

Proof of Theorem elsuppfn
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 suppvalfn 6802 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e.  X  |  ( F `
 i )  =/= 
Z } )
21eleq2d 2522 . 2  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( S  e.  ( F supp  Z )  <->  S  e.  { i  e.  X  | 
( F `  i
)  =/=  Z }
) )
3 fveq2 5794 . . . 4  |-  ( i  =  S  ->  ( F `  i )  =  ( F `  S ) )
43neeq1d 2726 . . 3  |-  ( i  =  S  ->  (
( F `  i
)  =/=  Z  <->  ( F `  S )  =/=  Z
) )
54elrab 3218 . 2  |-  ( S  e.  { i  e.  X  |  ( F `
 i )  =/= 
Z }  <->  ( S  e.  X  /\  ( F `  S )  =/=  Z ) )
62, 5syl6bb 261 1  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( S  e.  ( F supp  Z )  <->  ( S  e.  X  /\  ( F `  S )  =/=  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   {crab 2800    Fn wfn 5516   ` cfv 5521  (class class class)co 6195   supp csupp 6795
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-supp 6796
This theorem is referenced by:  fvn0elsupp  6811  rexsupp  6812  suppss  6824  suppssr  6825  wemapso2lem  7873  cantnfle  7985  cantnfp1lem2  7993  cantnfp1lem3  7994  cantnfp1  7995  cantnflem1a  7999  cantnflem3  8005  cnfcomlem  8038  cnfcom3  8043  mdegleb  21663  suppssfz  30929
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