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Theorem suppvalfn 6810
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 5619 . . . 4  |-  ( F  Fn  X  ->  Fun  F )
213ad2ant1 1009 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  Fun  F )
3 fnex 6056 . . . 4  |-  ( ( F  Fn  X  /\  X  e.  V )  ->  F  e.  _V )
433adant3 1008 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  F  e.  _V )
5 simp3 990 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  Z  e.  W )
6 suppval1 6809 . . 3  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
i  e.  dom  F  |  ( F `  i )  =/=  Z } )
72, 4, 5, 6syl3anc 1219 . 2  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  { i  e. 
dom  F  |  ( F `  i )  =/=  Z } )
8 fndm 5621 . . . 4  |-  ( F  Fn  X  ->  dom  F  =  X )
983ad2ant1 1009 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  dom  F  =  X )
10 rabeq 3072 . . 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 2495 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078   dom cdm 4951   Fun wfun 5523    Fn wfn 5524   ` cfv 5529  (class class class)co 6203   supp csupp 6803
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-supp 6804
This theorem is referenced by:  elsuppfn  6811  cantnflem1  8012  mptscmfsupp0  17144  rrgsupp  17495  frlmbas  18315  frlmssuvc2  18355  rrxmvallem  21045  fpwrelmapffslem  26210  fsumcvg4  26548  fsuppmapnn0ub  30967  fsuppmapnn0fiub0  30972  mptnn0fsupp  30973  mptnn0fsuppr  30975  pmatcollpw2lem  31287
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