MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgraedg2 Structured version   Unicode version

Theorem usgraedg2 21395
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 21356. (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 21375 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 f1f 5639 . . . 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 5870 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
5 fveq2 5728 . . . . 5  |-  ( x  =  ( E `  X )  ->  ( # `
 x )  =  ( # `  ( E `  X )
) )
65eqeq1d 2444 . . . 4  |-  ( x  =  ( E `  X )  ->  (
( # `  x )  =  2  <->  ( # `  ( E `  X )
)  =  2 ) )
76elrab 3092 . . 3  |-  ( ( E `  X )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( ( E `  X )  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 ( E `  X ) )  =  2 ) )
87simprbi 451 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    \ cdif 3317   (/)c0 3628   ~Pcpw 3799   {csn 3814   class class class wbr 4212   dom cdm 4878   -->wf 5450   -1-1->wf1 5451   ` cfv 5454   2c2 10049   #chash 11618   USGrph cusg 21365
This theorem is referenced by:  usgraedgprv  21396  usgranloopv  21398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fv 5462  df-usgra 21367
  Copyright terms: Public domain W3C validator