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Theorem usgrafilem1 25218
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 3569 . 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 3758 . . 3  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )
3 df-nel 2644 . . . . . . 7  |-  ( N  e/  ( E `  x )  <->  -.  N  e.  ( E `  x
) )
43bicomi 207 . . . . . 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 3019 . . . 4  |-  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) }  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
76uneq2i 3576 . . 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 2493 . 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 5041 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
11 ssrab2 3500 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  C_  dom  E
12 ssdmres 5132 . . . . 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 213 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1410, 13eqtri 2493 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1514uneq1i 3575 . 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 2503 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 189    = wceq 1452    e. wcel 1904    e/ wnel 2642   {crab 2760    u. cun 3388    C_ wss 3390   dom cdm 4839    |` cres 4841   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-dm 4849  df-res 4851
This theorem is referenced by:  usgrafilem2  25219  cusgrasizeindslem1  25280
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