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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 USGrph
Distinct variable groups:   ,   ,

Proof of Theorem usgraf
StepHypRef Expression
1 usgrav 24465 . . 3 USGrph
2 isusgra 24471 . . 3 USGrph
31, 2syl 16 . 2 USGrph USGrph
43ibi 241 1 USGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  crab 2811  cvv 3109   cdif 3468  c0 3793  cpw 4015  csn 4032   class class class wbr 4456   cdm 5008  wf1 5591  cfv 5594  c2 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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