Theorem uspgrupgrushgr 16036

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 16035	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
2		uspgrushgr 16034	. . 3 |- ( G e. USPGraph -> G e. USHGraph )
3	1, 2	jca 306	. 2 |- ( G e. USPGraph -> ( G e. UPGraph /\ G e. USHGraph ) )
4		eqid 2231	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2231	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6	4, 5	ushgrfm 15928	. . . 4 |- ( G e. USHGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | E. y y e. x } )
7		edgvalg 15913	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8		upgredgssen 15993	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
9	7, 8	eqsstrrd 3264	. . . 4 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
10		f1ssr 5549	. . . 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 16011	. . . 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 715   E.wex 1540   e. wcel 2202   {crab 2514   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   ran crn 4726   -1-1->wf1 5323   ` cfv 5326   1oc1o 6575   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15866  iEdgciedg 15867  Edgcedg 15911  USHGraphcushgr 15922  UPGraphcupgr 15945  USPGraphcuspgr 16007

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-ushgrm 15924  df-upgren 15947  df-uspgren 16009

This theorem is referenced by: (None)