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Theorem usgraedg2 27347
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 24157. (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 27327 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 f1f 5475 . . . 4  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
31, 2syl 15 . . 3  |-  ( V USGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
43ffvelrnda 5703 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
5 fveq2 5563 . . . . 5  |-  ( x  =  ( E `  X )  ->  ( # `
 x )  =  ( # `  ( E `  X )
) )
65eqeq1d 2324 . . . 4  |-  ( x  =  ( E `  X )  ->  (
( # `  x )  =  2  <->  ( # `  ( E `  X )
)  =  2 ) )
76elrab 2957 . . 3  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( ( E `  X )  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 ( E `  X ) )  =  2 ) )
87simprbi 450 . 2  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  ( # `  ( E `  X )
)  =  2 )
94, 8syl 15 1  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   {crab 2581    \ cdif 3183   (/)c0 3489   ~Pcpw 3659   {csn 3674   class class class wbr 4060   dom cdm 4726   -->wf 5288   -1-1->wf1 5289   ` cfv 5292   2c2 9840   #chash 11384   USGrph cusg 27317
This theorem is referenced by:  usgraedgprv  27348  usgranloop  27350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fv 5300  df-usgra 27319
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