Theorem umgrupgr 25998

Description: An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.)

Assertion

Ref	Expression
umgrupgr	|- ( G e. UMGraph -> G e. UPGraph )

Proof of Theorem umgrupgr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	isumgr 25990	. . . 4 |- ( G e. UMGraph -> ( G e. UMGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
4		id 22	. . . . 5 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
5		2re 11090	. . . . . . . . . . 11 |- 2 e. RR
6	5	leidi 10562	. . . . . . . . . 10 |- 2 <_ 2
7	6	a1i 11	. . . . . . . . 9 |- ( ( # ` x ) = 2 -> 2 <_ 2 )
8		breq1 4656	. . . . . . . . 9 |- ( ( # ` x ) = 2 -> ( ( # ` x ) <_ 2 <-> 2 <_ 2 ) )
9	7, 8	mpbird 247	. . . . . . . 8 |- ( ( # ` x ) = 2 -> ( # ` x ) <_ 2 )
10	9	a1i 11	. . . . . . 7 |- ( x e. ( ~P (Vtx ` G ) \ { (/) } ) -> ( ( # ` x ) = 2 -> ( # ` x ) <_ 2 ) )
11	10	ss2rabi 3684	. . . . . 6 |- { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 }
12	11	a1i 11	. . . . 5 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
13	4, 12	fssd 6057	. . . 4 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
14	3, 13	syl6bi 243	. . 3 |- ( G e. UMGraph -> ( G e. UMGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
15	14	pm2.43i 52	. 2 |- ( G e. UMGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
16	1, 2	isupgr 25979	. 2 |- ( G e. UMGraph -> ( G e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
17	15, 16	mpbird 247	1 |- ( G e. UMGraph -> G e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975   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-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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010

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-ne 2795  df-nel 2898  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-res 5126  df-ima 5127  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-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-2 11079  df-upgr 25977  df-umgr 25978

This theorem is referenced by:  umgruhgr  25999  upgr0e  26006  umgrislfupgr  26018  nbumgrvtx  26242  umgrwlknloop  26545  umgrwwlks2on  26850  umgr3v3e3cycl  27044  konigsberg  27119