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Theorem usgrafilem1 21378
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 3451 . 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 3610 . . 3  |-  dom  E  =  ( { x  e.  dom  E  |  N  e.  ( E `  x
) }  u.  {
x  e.  dom  E  |  -.  N  e.  ( E `  x ) } )
3 df-nel 2570 . . . . . . 7  |-  ( N  e/  ( E `  x )  <->  -.  N  e.  ( E `  x
) )
43bicomi 194 . . . . . 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 2906 . . . 4  |-  { x  e.  dom  E  |  -.  N  e.  ( E `  x ) }  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
76uneq2i 3458 . . 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 2424 . 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 5030 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
11 ssrab2 3388 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  C_  dom  E
12 ssdmres 5127 . . . . 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 200 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1410, 13eqtri 2424 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
1514uneq1i 3457 . 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 2434 1  |-  dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1721    e/ wnel 2568   {crab 2670    u. cun 3278    C_ wss 3280   dom cdm 4837    |` cres 4839   ` cfv 5413
This theorem is referenced by:  usgrafilem2  21379  cusgrasizeindslem1  21435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-dm 4847  df-res 4849
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