Theorem usgrumgruspgr 16167

Description: A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)

Assertion

Ref	Expression
usgrumgruspgr	|- ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )

Proof of Theorem usgrumgruspgr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrumgr 16166	. . 3 |- ( G e. USGraph -> G e. UMGraph )
2		usgruspgr 16165	. . 3 |- ( G e. USGraph -> G e. USPGraph )
3	1, 2	jca 306	. 2 |- ( G e. USGraph -> ( G e. UMGraph /\ G e. USPGraph ) )
4		eqid 2232	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2232	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	uspgrfen 16141	. . . 4 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7		edgvalg 16041	. . . . 5 |- ( G e. UMGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8		umgredgssen 16122	. . . . 5 |- ( G e. UMGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
9	7, 8	eqsstrrd 3274	. . . 4 |- ( G e. UMGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
10		f1ssr 5579	. . . 4 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } /\ ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } )
11	6, 9, 10	syl2anr 290	. . 3 |- ( ( G e. UMGraph /\ G e. USPGraph ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } )
12	4, 5	isusgren 16140	. . . 4 |- ( G e. UMGraph -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
13	12	adantr 276	. . 3 |- ( ( G e. UMGraph /\ G e. USPGraph ) -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
14	11, 13	mpbird 167	. 2 |- ( ( G e. UMGraph /\ G e. USPGraph ) -> G e. USGraph )
15	3, 14	impbii 126	1 |- ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 716   e. wcel 2203   {crab 2524   C_ wss 3210   ~Pcpw 3668   class class class wbr 4108   dom cdm 4748   ran crn 4749   -1-1->wf1 5348   ` cfv 5351   1oc1o 6639   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15994  iEdgciedg 15995  Edgcedg 16039  UMGraphcumgr 16074  USPGraphcuspgr 16135  USGraphcusgr 16136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-umgren 16076  df-uspgren 16137  df-usgren 16138

This theorem is referenced by: (None)