Theorem usgrumgruspgr 16197

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 16196	. . 3 |- ( G e. USGraph -> G e. UMGraph )
2		usgruspgr 16195	. . 3 |- ( G e. USGraph -> G e. USPGraph )
3	1, 2	jca 306	. 2 |- ( G e. USGraph -> ( G e. UMGraph /\ G e. USPGraph ) )
4		eqid 2234	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2234	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	uspgrfen 16171	. . . 4 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
7		edgvalg 16071	. . . . 5 |- ( G e. UMGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8		umgredgssen 16152	. . . . 5 |- ( G e. UMGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
9	7, 8	eqsstrrd 3277	. . . 4 |- ( G e. UMGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
10		f1ssr 5582	. . . 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 16170	. . . 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 2205   {crab 2526   C_ wss 3213   ~Pcpw 3671   class class class wbr 4111   dom cdm 4751   ran crn 4752   -1-1->wf1 5351   ` cfv 5354   1oc1o 6642   2oc2o 6643   ~~ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  Edgcedg 16069  UMGraphcumgr 16104  USPGraphcuspgr 16165  USGraphcusgr 16166

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-umgren 16106  df-uspgren 16167  df-usgren 16168

This theorem is referenced by: (None)