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Theorem usgraedg2 28132
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 23875. (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 28116 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 f1f 5439 . . . 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 5667 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
5 fveq2 5527 . . . . 5  |-  ( x  =  ( E `  X )  ->  ( # `
 x )  =  ( # `  ( E `  X )
) )
65eqeq1d 2293 . . . 4  |-  ( x  =  ( E `  X )  ->  (
( # `  x )  =  2  <->  ( # `  ( E `  X )
)  =  2 ) )
76elrab 2925 . . 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 1625    e. wcel 1686   {crab 2549    \ cdif 3151   (/)c0 3457   ~Pcpw 3627   {csn 3642   class class class wbr 4025   dom cdm 4691   -->wf 5253   -1-1->wf1 5254   ` cfv 5257   2c2 9797   #chash 11339   USGrph cusg 28106
This theorem is referenced by:  usgraedgprv  28133  usgranloop  28135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fv 5265  df-usgra 28108
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