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Theorem suppval 6923
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 6922 . . 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 5050 . . . . 5  |-  ( x  =  X  ->  dom  x  =  dom  X )
43adantr 466 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  dom  x  =  dom  X )
5 imaeq1 5178 . . . . . 6  |-  ( x  =  X  ->  (
x " { i } )  =  ( X " { i } ) )
65adantr 466 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( x " {
i } )  =  ( X " {
i } ) )
7 sneq 4006 . . . . . 6  |-  ( z  =  Z  ->  { z }  =  { Z } )
87adantl 467 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  { z }  =  { Z } )
96, 8neeq12d 2703 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( ( x " { i } )  =/=  { z }  <-> 
( X " {
i } )  =/= 
{ Z } ) )
104, 9rabeqbidv 3076 . . 3  |-  ( ( x  =  X  /\  z  =  Z )  ->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } }  =  { i  e.  dom  X  | 
( X " {
i } )  =/= 
{ Z } }
)
1110adantl 467 . 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 3090 . . 3  |-  ( X  e.  V  ->  X  e.  _V )
1312adantr 466 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  X  e.  _V )
14 elex 3090 . . 3  |-  ( Z  e.  W  ->  Z  e.  _V )
1514adantl 467 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  Z  e.  _V )
16 dmexg 6734 . . . 4  |-  ( X  e.  V  ->  dom  X  e.  _V )
1716adantr 466 . . 3  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  dom  X  e.  _V )
18 rabexg 4570 . . 3  |-  ( dom 
X  e.  _V  ->  { i  e.  dom  X  |  ( X " { i } )  =/=  { Z } }  e.  _V )
1917, 18syl 17 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  { i  e.  dom  X  |  ( X " { i } )  =/=  { Z } }  e.  _V )
202, 11, 13, 15, 19ovmpt2d 6434 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 370    = wceq 1437    e. wcel 1868    =/= wne 2618   {crab 2779   _Vcvv 3081   {csn 3996   dom cdm 4849   "cima 4852  (class class class)co 6301    |-> cmpt2 6303   supp csupp 6921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-supp 6922
This theorem is referenced by:  suppvalbr  6925  supp0  6926  suppval1  6927  suppssdm  6934  suppsnop  6935  ressuppss  6941  ressuppssdif  6943
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