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Theorem suppvalfn 6824
Description: The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
Assertion
Ref Expression
suppvalfn  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e.  X  |  ( F `
 i )  =/= 
Z } )
Distinct variable groups:    i, V    i, W    i, X    i, Z    i, F

Proof of Theorem suppvalfn
StepHypRef Expression
1 fnfun 5586 . . . 4  |-  ( F  Fn  X  ->  Fun  F )
213ad2ant1 1015 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  Fun  F )
3 fnex 6040 . . . 4  |-  ( ( F  Fn  X  /\  X  e.  V )  ->  F  e.  _V )
433adant3 1014 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  F  e.  _V )
5 simp3 996 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  Z  e.  W )
6 suppval1 6823 . . 3  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
i  e.  dom  F  |  ( F `  i )  =/=  Z } )
72, 4, 5, 6syl3anc 1226 . 2  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e. 
dom  F  |  ( F `  i )  =/=  Z } )
8 fndm 5588 . . . 4  |-  ( F  Fn  X  ->  dom  F  =  X )
983ad2ant1 1015 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  dom  F  =  X )
10 rabeq 3028 . . 3  |-  ( dom 
F  =  X  ->  { i  e.  dom  F  |  ( F `  i )  =/=  Z }  =  { i  e.  X  |  ( F `  i )  =/=  Z } )
119, 10syl 16 . 2  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  { i  e.  dom  F  |  ( F `  i )  =/=  Z }  =  { i  e.  X  |  ( F `  i )  =/=  Z } )
127, 11eqtrd 2423 1  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e.  X  |  ( F `
 i )  =/= 
Z } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   {crab 2736   _Vcvv 3034   dom cdm 4913   Fun wfun 5490    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:  elsuppfn  6825  cantnflem1  8021  fsuppmapnn0fiub0  12002  fsuppmapnn0ub  12004  mptnn0fsupp  12006  mptnn0fsuppr  12008  cicer  15212  mptscmfsupp0  17689  rrgsupp  18052  frlmbas  18877  frlmssuvc2  18915  pmatcollpw2lem  19363  rrxmvallem  21916  fpwrelmapffslem  27705  fsumcvg4  28086
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