Theorem usgrf1oedg 26099

Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgrf1oedg.i	|- I = (iEdg ` G )
usgrf1oedg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
usgrf1oedg	|- ( G e. USGraph -> I : dom I -1-1-onto-> E )

Proof of Theorem usgrf1oedg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		usgrf1oedg.i	. . . 4 |- I = (iEdg ` G )
3	1, 2	usgrf 26050	. . 3 |- ( G e. USGraph -> I : dom I -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4		f1f1orn 6148	. . 3 |- ( I : dom I -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> I : dom I -1-1-onto-> ran I )
5	3, 4	syl 17	. 2 |- ( G e. USGraph -> I : dom I -1-1-onto-> ran I )
6		usgrf1oedg.e	. . . 4 |- E = (Edg ` G )
7		edgval 25941	. . . . . 6 |- (Edg ` G ) = ran (iEdg ` G )
8	7	a1i 11	. . . . 5 |- ( G e. USGraph -> (Edg ` G ) = ran (iEdg ` G ) )
9	2	eqcomi 2631	. . . . . 6 |- (iEdg ` G ) = I
10	9	rneqi 5352	. . . . 5 |- ran (iEdg ` G ) = ran I
11	8, 10	syl6eq 2672	. . . 4 |- ( G e. USGraph -> (Edg ` G ) = ran I )
12	6, 11	syl5eq 2668	. . 3 |- ( G e. USGraph -> E = ran I )
13	12	f1oeq3d 6134	. 2 |- ( G e. USGraph -> ( I : dom I -1-1-onto-> E <-> I : dom I -1-1-onto-> ran I ) )
14	5, 13	mpbird 247	1 |- ( G e. USGraph -> I : dom I -1-1-onto-> E )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   ran crn 5115   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-mo 2475  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-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  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-fo 5894  df-f1o 5895  df-fv 5896  df-edg 25940  df-usgr 26046

This theorem is referenced by:  usgr2trlncl  26656