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Theorem usgrafilem1 25124
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 3610 . 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 3785 . . 3  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )
3 df-nel 2621 . . . . . . 7  |-  ( N  e/  ( E `  x )  <->  -.  N  e.  ( E `  x
) )
43bicomi 205 . . . . . 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 3069 . . . 4  |-  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) }  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
76uneq2i 3617 . . 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 2451 . 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 5051 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
11 ssrab2 3546 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  C_  dom  E
12 ssdmres 5141 . . . . 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 211 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1410, 13eqtri 2451 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1514uneq1i 3616 . 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 2461 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 187    = wceq 1437    e. wcel 1868    e/ wnel 2619   {crab 2779    u. cun 3434    C_ wss 3436   dom cdm 4849    |` cres 4851   ` cfv 5597
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-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
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-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  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-br 4421  df-opab 4480  df-xp 4855  df-dm 4859  df-res 4861
This theorem is referenced by:  usgrafilem2  25125  cusgrasizeindslem1  25186
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