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Theorem suppval 6803
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 6802 . . 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 5149 . . . . 5  |-  ( x  =  X  ->  dom  x  =  dom  X )
43adantr 465 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  dom  x  =  dom  X )
5 imaeq1 5273 . . . . . 6  |-  ( x  =  X  ->  (
x " { i } )  =  ( X " { i } ) )
65adantr 465 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( x " {
i } )  =  ( X " {
i } ) )
7 sneq 3996 . . . . . 6  |-  ( z  =  Z  ->  { z }  =  { Z } )
87adantl 466 . . . . 5  |-  ( ( x  =  X  /\  z  =  Z )  ->  { z }  =  { Z } )
96, 8neeq12d 2731 . . . 4  |-  ( ( x  =  X  /\  z  =  Z )  ->  ( ( x " { i } )  =/=  { z }  <-> 
( X " {
i } )  =/= 
{ Z } ) )
104, 9rabeqbidv 3073 . . 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 3087 . . 3  |-  ( X  e.  V  ->  X  e.  _V )
1312adantr 465 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  X  e.  _V )
14 elex 3087 . . 3  |-  ( Z  e.  W  ->  Z  e.  _V )
1514adantl 466 . 2  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  Z  e.  _V )
16 dmexg 6620 . . . 4  |-  ( X  e.  V  ->  dom  X  e.  _V )
1716adantr 465 . . 3  |-  ( ( X  e.  V  /\  Z  e.  W )  ->  dom  X  e.  _V )
18 rabexg 4551 . . 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 6329 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 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078   {csn 3986   dom cdm 4949   "cima 4952  (class class class)co 6201    |-> cmpt2 6203   supp csupp 6801
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-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-supp 6802
This theorem is referenced by:  suppvalbr  6805  supp0  6806  suppval1  6807  suppssdm  6814  suppsnop  6815  ressuppss  6819  ressuppssdif  6821
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