Theorem uspgrupgrushgr 16423

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 16422	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
2		uspgrushgr 16421	. . 3 |- ( G e. USPGraph -> G e. USHGraph )
3	1, 2	jca 306	. 2 |- ( G e. USPGraph -> ( G e. UPGraph /\ G e. USHGraph ) )
4		eqid 2238	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2238	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	ushgrfm 16315	. . . 4 |- ( G e. USHGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | E. y y e. x } )
7		edgvalg 16300	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8		upgredgssen 16380	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
9	7, 8	eqsstrrd 3285	. . . 4 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
10		f1ssr 5605	. . . 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 16398	. . . 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 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   \/ wo 720   E.wex 1545   e. wcel 2209   {crab 2532   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   dom cdm 4774   ran crn 4775   -1-1->wf1 5374   ` cfv 5377   1oc1o 6680   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  USHGraphcushgr 16309  UPGraphcupgr 16332  USPGraphcuspgr 16394

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-ushgrm 16311  df-upgren 16334  df-uspgren 16396

This theorem is used by: (None)