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Theorem usgraf 24473
Description: The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
usgraf  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
Distinct variable groups:    x, E    x, V

Proof of Theorem usgraf
StepHypRef Expression
1 usgrav 24465 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 isusgra 24471 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
31, 2syl 16 . 2  |-  ( V USGrph  E  ->  ( V USGrph  E  <->  E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } ) )
43ibi 241 1  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   dom cdm 5008   -1-1->wf1 5591   ` cfv 5594   2c2 10606   #chash 12408   USGrph cusg 24457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-usgra 24460
This theorem is referenced by:  usgrares  24496  usgraedg2  24502  constr3trllem1  24777  constr3trllem5  24781  usgedgffibi  32696
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