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Theorem elsuppfn 6825
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 6824 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e.  X  |  ( F `
 i )  =/= 
Z } )
21eleq2d 2452 . 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 5774 . . . 4  |-  ( i  =  S  ->  ( F `  i )  =  ( F `  S ) )
43neeq1d 2659 . . 3  |-  ( i  =  S  ->  (
( F `  i
)  =/=  Z  <->  ( F `  S )  =/=  Z
) )
54elrab 3182 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   {crab 2736    Fn wfn 5491   ` cfv 5496  (class class class)co 6196   supp csupp 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-supp 6818
This theorem is referenced by:  fvn0elsupp  6833  fvn0elsuppOLD  6834  fvn0elsuppb  6835  rexsupp  6836  suppss  6848  suppssr  6849  wemapso2lem  7893  cantnfle  8003  cantnfp1lem2  8011  cantnfp1lem3  8012  cantnfp1  8013  cantnflem1a  8017  cantnflem3  8023  cnfcomlem  8056  cnfcom3  8061  suppssfz  12003  ciclcl  15208  cicrcl  15209  mdegleb  22549  fdivmptf  33362  refdivmptf  33363
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