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Theorem usgraedg2 24779
Description: The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 24725. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
Assertion
Ref Expression
usgraedg2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )

Proof of Theorem usgraedg2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgraf 24750 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 f1f 5763 . . . 4  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
31, 2syl 17 . . 3  |-  ( V USGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
43ffvelrnda 6008 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
5 fveq2 5848 . . . . 5  |-  ( x  =  ( E `  X )  ->  ( # `
 x )  =  ( # `  ( E `  X )
) )
65eqeq1d 2404 . . . 4  |-  ( x  =  ( E `  X )  ->  (
( # `  x )  =  2  <->  ( # `  ( E `  X )
)  =  2 ) )
76elrab 3206 . . 3  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( ( E `  X )  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 ( E `  X ) )  =  2 ) )
87simprbi 462 . 2  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  ( # `  ( E `  X )
)  =  2 )
94, 8syl 17 1  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2757    \ cdif 3410   (/)c0 3737   ~Pcpw 3954   {csn 3971   class class class wbr 4394   dom cdm 4822   -->wf 5564   -1-1->wf1 5565   ` cfv 5568   2c2 10625   #chash 12450   USGrph cusg 24734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fv 5576  df-usgra 24737
This theorem is referenced by:  usgraedgprv  24780  usgranloopv  24782
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