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Theorem suppval 6919
Description: The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
Assertion
Ref Expression
suppval  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  ( X supp  Z )  =  { i  e. 
dom  X  |  ( X " { i } )  =/=  { Z } } )
Distinct variable groups:    i, X    i, Z
Allowed substitution hints:    V( i)    W( i)

Proof of Theorem suppval
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-supp 6918 . . 3  |- supp  =  ( x  e.  _V , 
z  e.  _V  |->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } } )
21a1i 11 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  -> supp  =  ( x  e. 
_V ,  z  e. 
_V  |->  { i  e. 
dom  x  |  ( x " { i } )  =/=  {
z } } ) )
3 dmeq 5213 . . . . 5  |-  ( x  =  X  ->  dom  x  =  dom  X )
43adantr 465 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  dom  x  =  dom  X )
5 imaeq1 5342 . . . . . 6  |-  ( x  =  X  ->  (
x " { i } )  =  ( X " { i } ) )
65adantr 465 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( x " {
i } )  =  ( X " {
i } ) )
7 sneq 4042 . . . . . 6  |-  ( z  =  Z  ->  { z }  =  { Z } )
87adantl 466 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  { z }  =  { Z } )
96, 8neeq12d 2736 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( ( x " { i } )  =/=  { z }  <-> 
( X " {
i } )  =/= 
{ Z } ) )
104, 9rabeqbidv 3104 . . 3  |-  ( ( x  =  X  /\  z  =  Z )  ->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } }  =  { i  e.  dom  X  | 
( X " {
i } )  =/= 
{ Z } }
)
1110adantl 466 . 2  |-  ( ( ( X  e.  V  /\  Z  e.  W
)  /\  ( x  =  X  /\  z  =  Z ) )  ->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } }  =  { i  e.  dom  X  | 
( X " {
i } )  =/= 
{ Z } }
)
12 elex 3118 . . 3  |-  ( X  e.  V  ->  X  e.  _V )
1312adantr 465 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  X  e.  _V )
14 elex 3118 . . 3  |-  ( Z  e.  W  ->  Z  e.  _V )
1514adantl 466 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  Z  e.  _V )
16 dmexg 6730 . . . 4  |-  ( X  e.  V  ->  dom  X  e.  _V )
1716adantr 465 . . 3  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  dom  X  e.  _V )
18 rabexg 4606 . . 3  |-  ( dom 
X  e.  _V  ->  { i  e.  dom  X  |  ( X " { i } )  =/=  { Z } }  e.  _V )
1917, 18syl 16 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  { i  e.  dom  X  |  ( X " { i } )  =/=  { Z } }  e.  _V )
202, 11, 13, 15, 19ovmpt2d 6429 1  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  ( X supp  Z )  =  { i  e. 
dom  X  |  ( X " { i } )  =/=  { Z } } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   {csn 4032   dom cdm 5008   "cima 5011  (class class class)co 6296    |-> cmpt2 6298   supp csupp 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-supp 6918
This theorem is referenced by:  suppvalbr  6921  supp0  6922  suppval1  6923  suppssdm  6930  suppsnop  6931  ressuppss  6937  ressuppssdif  6939
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