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Theorem cusgrasizeindslem1 24772
Description: Lemma 1 for cusgrasizeinds 24775. 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
cusgrares.f  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
Assertion
Ref Expression
cusgrasizeindslem1  |-  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 cusgrasizeindslem1
StepHypRef Expression
1 cusgrares.f . 2  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
21usgrafilem1 24710 1  |-  dom  E  =  ( dom  F  u.  { x  e.  dom  E  |  N  e.  ( E `  x ) } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403    e. wcel 1840    e/ wnel 2597   {crab 2755    u. cun 3409   dom cdm 4940    |` cres 4942   ` cfv 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946  df-dm 4950  df-res 4952
This theorem is referenced by:  cusgrasizeinds  24775
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