Theorem usgruspgrben 16310

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 16307	. . 3 |- ( G e. USGraph -> G e. USPGraph )
2		edgusgren 16287	. . . . 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 2617	. . 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 16183	. . . . . 6 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
7	6	raleqdv 2749	. . . . 5 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) e ~~ 2o <-> A. e e. ran (iEdg ` G ) e ~~ 2o ) )
8		eqid 2234	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
9		eqid 2234	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
10	8, 9	uspgrfen 16283	. . . . . 6 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } )
11		f1rn 5579	. . . . . . . . 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 3237	. . . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . . . 16 |- ( e = y -> ( e ~~ 2o <-> y ~~ 2o ) )
15	14	rspcv 2919	. . . . . . . . . . . . . . 15 |- ( y e. ran (iEdg ` G ) -> ( A. e e. ran (iEdg ` G ) e ~~ 2o -> y ~~ 2o ) )
16		breq1 4117	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 1o <-> y ~~ 1o ) )
17		breq1 4117	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( x ~~ 2o <-> y ~~ 2o ) )
18	16, 17	orbi12d 801	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( y ~~ 1o \/ y ~~ 2o ) ) )
19	18	elrab 2976	. . . . . . . . . . . . . . . 16 |- ( y e. { x e. ~P (Vtx ` G ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( y e. ~P (Vtx ` G ) /\ ( y ~~ 1o \/ y ~~ 2o ) ) )
20	17	elrab 2976	. . . . . . . . . . . . . . . . . 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 3248	. . . . . . . . . 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 5585	. . . . . . . 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 16282	. . . 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 716   e. wcel 2205   A.wral 2522   {crab 2526   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   ran crn 4755   -1-1->wf1 5354   ` cfv 5357   1oc1o 6653   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16136  iEdgciedg 16137  Edgcedg 16181  USPGraphcuspgr 16277  USGraphcusgr 16278

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-edg 16182  df-uspgren 16279  df-usgren 16280

This theorem is referenced by:  usgr1e  16365