Theorem umgr0e 26005

Description: The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
umgr0e.g	|- ( ph -> G e. W )
umgr0e.e	|- ( ph -> (iEdg ` G ) = (/) )

Assertion

Ref	Expression
umgr0e	|- ( ph -> G e. UMGraph )

Proof of Theorem umgr0e

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgr0e.e	. . . 4 |- ( ph -> (iEdg ` G ) = (/) )
2	1	f10d 6170	. . 3 |- ( ph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
3		f1f 6101	. . 3 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4	2, 3	syl 17	. 2 |- ( ph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
5		umgr0e.g	. . 3 |- ( ph -> G e. W )
6		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
7		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
8	6, 7	isumgr 25990	. . 3 |- ( G e. W -> ( G e. UMGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
9	5, 8	syl 17	. 2 |- ( ph -> ( G e. UMGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
10	4, 9	mpbird 247	1 |- ( ph -> G e. UMGraph )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UMGraph cumgr 25976

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-sep 4781  ax-nul 4789  ax-pr 4906

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-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-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-fv 5896  df-umgr 25978

This theorem is referenced by:  upgr0e  26006