Theorem usgrf 26050

Description: The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Hypotheses

Ref	Expression
isuspgr.v	|- V = (Vtx ` G )
isuspgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
usgrf	|- ( G e. USGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )

Distinct variable groups:   x, G   x, V

Allowed substitution hint:   E( x)

Proof of Theorem usgrf

Step	Hyp	Ref	Expression
1		isuspgr.v	. . 3 |- V = (Vtx ` G )
2		isuspgr.e	. . 3 |- E = (iEdg ` G )
3	1, 2	isusgr 26048	. 2 |- ( G e. USGraph -> ( G e. USGraph <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) )
4	3	ibi 256	1 |- ( G e. USGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   -1-1->wf1 5885   ` cfv 5888   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-usgr 26046

This theorem is referenced by:  usgredg2ALT  26085  usgrf1oedg  26099  usgrsizedg  26107  usgrres  26200  clwlksfclwwlk  26962