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Theorem usgrafilem1 24537
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 3644 . 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 3817 . . 3  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )
3 df-nel 2655 . . . . . . 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 3098 . . . 4  |-  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) }  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
76uneq2i 3651 . . 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 2486 . 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 5214 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
11 ssrab2 3581 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  C_  dom  E
12 ssdmres 5305 . . . . 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 2486 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1514uneq1i 3650 . 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 2496 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 1395    e. wcel 1819    e/ wnel 2653   {crab 2811    u. cun 3469    C_ wss 3471   dom cdm 5008    |` cres 5010   ` cfv 5594
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-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
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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  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-br 4457  df-opab 4516  df-xp 5014  df-dm 5018  df-res 5020
This theorem is referenced by:  usgrafilem2  24538  cusgrasizeindslem1  24599
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