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Theorem usgrafilem1 23243
Description: The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
Hypothesis
Ref Expression
usgrafis.f  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
Assertion
Ref Expression
usgrafilem1  |-  dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
Distinct variable groups:    x, E    x, N
Allowed substitution hint:    F( x)

Proof of Theorem usgrafilem1
StepHypRef Expression
1 uncom 3497 . 2  |-  ( { x  e.  dom  E  |  N  e.  ( E `  x ) }  u.  { x  e.  dom  E  |  N  e/  ( E `  x
) } )  =  ( { x  e. 
dom  E  |  N  e/  ( E `  x
) }  u.  {
x  e.  dom  E  |  N  e.  ( E `  x ) } )
2 rabxm 3657 . . 3  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )
3 df-nel 2607 . . . . . . 7  |-  ( N  e/  ( E `  x )  <->  -.  N  e.  ( E `  x
) )
43bicomi 202 . . . . . 6  |-  ( -.  N  e.  ( E `
 x )  <->  N  e/  ( E `  x ) )
54a1i 11 . . . . 5  |-  ( x  e.  dom  E  -> 
( -.  N  e.  ( E `  x
)  <->  N  e/  ( E `  x )
) )
65rabbiia 2959 . . . 4  |-  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) }  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
76uneq2i 3504 . . 3  |-  ( { x  e.  dom  E  |  N  e.  ( E `  x ) }  u.  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  N  e/  ( E `  x ) } )
82, 7eqtri 2461 . 2  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  N  e/  ( E `  x ) } )
9 usgrafis.f . . . . 5  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
109dmeqi 5037 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
11 ssrab2 3434 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  C_  dom  E
12 ssdmres 5129 . . . . 5  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  C_  dom  E  <->  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )  =  {
x  e.  dom  E  |  N  e/  ( E `  x ) } )
1311, 12mpbi 208 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1410, 13eqtri 2461 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1514uneq1i 3503 . 2  |-  ( dom 
F  u.  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  =  ( { x  e. 
dom  E  |  N  e/  ( E `  x
) }  u.  {
x  e.  dom  E  |  N  e.  ( E `  x ) } )
161, 8, 153eqtr4i 2471 1  |-  dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1364    e. wcel 1761    e/ wnel 2605   {crab 2717    u. cun 3323    C_ wss 3325   dom cdm 4836    |` cres 4838   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-dm 4846  df-res 4848
This theorem is referenced by:  usgrafilem2  23244  cusgrasizeindslem1  23300
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