Theorem uspgrupgrushgr 15980

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

Assertion

Ref	Expression
uspgrupgrushgr	|- ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )

Proof of Theorem uspgrupgrushgr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 15979	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
2		uspgrushgr 15978	. . 3 |- ( G e. USPGraph -> G e. USHGraph )
3	1, 2	jca 306	. 2 |- ( G e. USPGraph -> ( G e. UPGraph /\ G e. USHGraph ) )
4		eqid 2229	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2229	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	ushgrfm 15874	. . . 4 |- ( G e. USHGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | E. y y e. x } )
7		edgvalg 15860	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8		upgredgssen 15937	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
9	7, 8	eqsstrrd 3261	. . . 4 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
10		f1ssr 5538	. . . 4 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | E. y y e. x } /\ ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11	6, 9, 10	syl2anr 290	. . 3 |- ( ( G e. UPGraph /\ G e. USHGraph ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
12	4, 5	isuspgren 15955	. . . 4 |- ( G e. UPGraph -> ( G e. USPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
13	12	adantr 276	. . 3 |- ( ( G e. UPGraph /\ G e. USHGraph ) -> ( G e. USPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
14	11, 13	mpbird 167	. 2 |- ( ( G e. UPGraph /\ G e. USHGraph ) -> G e. USPGraph )
15	3, 14	impbii 126	1 |- ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 713   E.wex 1538   e. wcel 2200   {crab 2512   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   dom cdm 4719   ran crn 4720   -1-1->wf1 5315   ` cfv 5318   1oc1o 6555   2oc2o 6556   ~~ cen 6885  Vtxcvtx 15813  iEdgciedg 15814  Edgcedg 15858  USHGraphcushgr 15868  UPGraphcupgr 15891  USPGraphcuspgr 15951

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-1o 6562  df-2o 6563  df-en 6888  df-sub 8319  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-dec 9579  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-ushgrm 15870  df-upgren 15893  df-uspgren 15953

This theorem is referenced by: (None)