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Theorem usgraedg2 24037
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 23984. (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 24009 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 f1f 5772 . . . 4  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
31, 2syl 16 . . 3  |-  ( V USGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
43ffvelrnda 6012 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
5 fveq2 5857 . . . . 5  |-  ( x  =  ( E `  X )  ->  ( # `
 x )  =  ( # `  ( E `  X )
) )
65eqeq1d 2462 . . . 4  |-  ( x  =  ( E `  X )  ->  (
( # `  x )  =  2  <->  ( # `  ( E `  X )
)  =  2 ) )
76elrab 3254 . . 3  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( ( E `  X )  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 ( E `  X ) )  =  2 ) )
87simprbi 464 . 2  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  ( # `  ( E `  X )
)  =  2 )
94, 8syl 16 1  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811    \ cdif 3466   (/)c0 3778   ~Pcpw 4003   {csn 4020   class class class wbr 4440   dom cdm 4992   -->wf 5575   -1-1->wf1 5576   ` cfv 5579   2c2 10574   #chash 12360   USGrph cusg 23993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fv 5587  df-usgra 23996
This theorem is referenced by:  usgraedgprv  24038  usgranloopv  24040
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