Theorem usgruspgrben 16439

Description: A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)

Assertion

Ref	Expression
usgruspgrben	|- ( G e. USGraph <-> ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) )

Distinct variable group:   e, G

Proof of Theorem usgruspgrben

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

Step	Hyp	Ref	Expression
1		usgruspgr 16436	. . 3 |- ( G e. USGraph -> G e. USPGraph )
2		edgusgren 16416	. . . . 5 |- ( ( G e. USGraph /\ e e. (Edg ` G ) ) -> ( e e. ~P (Vtx ` G ) /\ e ~~ 2o ) )
3	2	simprd 114	. . . 4 |- ( ( G e. USGraph /\ e e. (Edg ` G ) ) -> e ~~ 2o )
4	3	ralrimiva 2623	. . 3 |- ( G e. USGraph -> A. e e. (Edg ` G ) e ~~ 2o )
5	1, 4	jca 306	. 2 |- ( G e. USGraph -> ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) )
6		edgvalg 16312	. . . . . 6 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
7	6	raleqdv 2755	. . . . 5 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) e ~~ 2o <-> A. e e. ran (iEdg ` G ) e ~~ 2o ) )
8		eqid 2238	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
9		eqid 2238	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
10	8, 9	uspgrfen 16412	. . . . . 6 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11		f1rn 5599	. . . . . . . . 9 |- ( (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 ~~ 1o \/ x ~~ 2o ) } )
12		ssel2 3243	. . . . . . . . . . . . . . 15 |- ( ( ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } /\ y e. ran (iEdg ` G ) ) -> y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
13	12	expcom 116	. . . . . . . . . . . . . 14 |- ( y e. ran (iEdg ` G ) -> ( ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
14		breq1 4133	. . . . . . . . . . . . . . . 16 |- ( e = y -> ( e ~~ 2o <-> y ~~ 2o ) )
15	14	rspcv 2925	. . . . . . . . . . . . . . 15 |- ( y e. ran (iEdg ` G ) -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> y ~~ 2o ) )
16		breq1 4133	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 1o <-> y ~~ 1o ) )
17		breq1 4133	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 2o <-> y ~~ 2o ) )
18	16, 17	orbi12d 805	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( y ~~ 1o \/ y ~~ 2o ) ) )
19	18	elrab 2982	. . . . . . . . . . . . . . . 16 |- ( y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( y e. ~P (Vtx ` G ) /\ ( y ~~ 1o \/ y ~~ 2o ) ) )
20	17	elrab 2982	. . . . . . . . . . . . . . . . . 18 |- ( y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } <-> ( y e. ~P (Vtx ` G ) /\ y ~~ 2o ) )
21	20	simplbi2 385	. . . . . . . . . . . . . . . . 17 |- ( y e. ~P (Vtx ` G ) -> ( y ~~ 2o -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
22	21	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( y e. ~P (Vtx ` G ) /\ ( y ~~ 1o \/ y ~~ 2o ) ) -> ( y ~~ 2o -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
23	19, 22	sylbi 121	. . . . . . . . . . . . . . 15 |- ( y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ( y ~~ 2o -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
24	15, 23	syl9 72	. . . . . . . . . . . . . 14 |- ( y e. ran (iEdg ` G ) -> ( y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) ) )
25	13, 24	syld 45	. . . . . . . . . . . . 13 |- ( y e. ran (iEdg ` G ) -> ( ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) ) )
26	25	com13 80	. . . . . . . . . . . 12 |- ( A. e e. ran (iEdg ` G ) e ~~ 2o -> ( ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ( y e. ran (iEdg ` G ) -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) ) )
27	26	imp 124	. . . . . . . . . . 11 |- ( ( A. e e. ran (iEdg ` G ) e ~~ 2o /\ ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) -> ( y e. ran (iEdg ` G ) -> y e. { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
28	27	ssrdv 3254	. . . . . . . . . 10 |- ( ( A. e e. ran (iEdg ` G ) e ~~ 2o /\ ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } ) -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
29	28	ex 115	. . . . . . . . 9 |- ( A. e e. ran (iEdg ` G ) e ~~ 2o -> ( ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
30	11, 29	mpan9 281	. . . . . . . 8 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } /\ A. e e. ran (iEdg ` G ) e ~~ 2o ) -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | x ~~ 2o } )
31		f1ssr 5605	. . . . . . . 8 |- ( ( (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 } )
32	30, 31	syldan 282	. . . . . . 7 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } /\ A. e e. ran (iEdg ` G ) e ~~ 2o ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } )
33	32	ex 115	. . . . . 6 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
34	10, 33	syl 14	. . . . 5 |- ( G e. USPGraph -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
35	7, 34	sylbid 150	. . . 4 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) e ~~ 2o -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
36	35	imp 124	. . 3 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } )
37	8, 9	isusgren 16411	. . . 4 |- ( G e. USPGraph -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
38	37	adantr 276	. . 3 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | x ~~ 2o } ) )
39	36, 38	mpbird 167	. 2 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) -> G e. USGraph )
40	5, 39	impbii 126	1 |- ( G e. USGraph <-> ( G e. USPGraph /\ A. e e. (Edg ` G ) e ~~ 2o ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   e. wcel 2209   A.wral 2528   {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 16265  iEdgciedg 16266  Edgcedg 16310  USPGraphcuspgr 16406  USGraphcusgr 16407

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-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-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-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-dec 9780  df-ndx 13357  df-slot 13358  df-base 13360  df-edgf 16258  df-vtx 16267  df-iedg 16268  df-edg 16311  df-uspgren 16408  df-usgren 16409

This theorem is used by:  usgr1e  16494