Theorem usgruspgrben 15941

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 15938	. . 3 |- ( G e. USGraph -> G e. USPGraph )
2		edgusgren 15918	. . . . 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 2581	. . 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 15817	. . . . . 6 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
7	6	raleqdv 2712	. . . . 5 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) e ~~ 2o <-> A. e e. ran (iEdg ` G ) e ~~ 2o ) )
8		eqid 2207	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
9		eqid 2207	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
10	8, 9	uspgrfen 15914	. . . . . 6 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11		f1rn 5505	. . . . . . . . 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 3197	. . . . . . . . . . . . . . 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 4063	. . . . . . . . . . . . . . . 16 |- ( e = y -> ( e ~~ 2o <-> y ~~ 2o ) )
15	14	rspcv 2881	. . . . . . . . . . . . . . 15 |- ( y e. ran (iEdg ` G ) -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> y ~~ 2o ) )
16		breq1 4063	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 1o <-> y ~~ 1o ) )
17		breq1 4063	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 2o <-> y ~~ 2o ) )
18	16, 17	orbi12d 795	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( y ~~ 1o \/ y ~~ 2o ) ) )
19	18	elrab 2937	. . . . . . . . . . . . . . . 16 |- ( y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( y e. ~P (Vtx ` G ) /\ ( y ~~ 1o \/ y ~~ 2o ) ) )
20	17	elrab 2937	. . . . . . . . . . . . . . . . . 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 3208	. . . . . . . . . 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 5511	. . . . . . . 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 15913	. . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   e. wcel 2178   A.wral 2486   {crab 2490   C_ wss 3175   ~Pcpw 3627   class class class wbr 4060   dom cdm 4694   ran crn 4695   -1-1->wf1 5288   ` cfv 5291   1oc1o 6520   2oc2o 6521   ~~ cen 6850  Vtxcvtx 15772  iEdgciedg 15773  Edgcedg 15815  USPGraphcuspgr 15908  USGraphcusgr 15909

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-cnre 8073

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-sub 8282  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-5 9135  df-6 9136  df-7 9137  df-8 9138  df-9 9139  df-n0 9333  df-dec 9542  df-ndx 12996  df-slot 12997  df-base 12999  df-edgf 15765  df-vtx 15774  df-iedg 15775  df-edg 15816  df-uspgren 15910  df-usgren 15911

This theorem is referenced by: (None)