Theorem usgruspgrb 26076

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

Assertion

Ref	Expression
usgruspgrb	|- ( G e. USGraph <-> ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) )

Distinct variable group:   e, G

Proof of Theorem usgruspgrb

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

Step	Hyp	Ref	Expression
1		usgruspgr 26073	. . 3 |- ( G e. USGraph -> G e. USPGraph )
2		edgusgr 26055	. . . . 5 |- ( ( G e. USGraph /\ e e. (Edg ` G ) ) -> ( e e. ~P (Vtx ` G ) /\ ( # ` e ) = 2 ) )
3	2	simprd 479	. . . 4 |- ( ( G e. USGraph /\ e e. (Edg ` G ) ) -> ( # ` e ) = 2 )
4	3	ralrimiva 2966	. . 3 |- ( G e. USGraph -> A. e e. (Edg ` G ) ( # ` e ) = 2 )
5	1, 4	jca 554	. 2 |- ( G e. USGraph -> ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) )
6		edgval 25941	. . . . . . 7 |- (Edg ` G ) = ran (iEdg ` G )
7	6	a1i 11	. . . . . 6 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
8	7	raleqdv 3144	. . . . 5 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) ( # ` e ) = 2 <-> A. e e. ran (iEdg ` G ) ( # ` e ) = 2 ) )
9		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
10		eqid 2622	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
11	9, 10	uspgrf 26049	. . . . . 6 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
12		f1f 6101	. . . . . . . . . 10 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
13		frn 6053	. . . . . . . . . 10 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
14	12, 13	syl 17	. . . . . . . . 9 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
15		ssel2 3598	. . . . . . . . . . . . . . 15 |- ( ( ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ y e. ran (iEdg ` G ) ) -> y e. { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
16	15	expcom 451	. . . . . . . . . . . . . 14 |- ( y e. ran (iEdg ` G ) -> ( ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> y e. { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
17		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( e = y -> ( # ` e ) = ( # ` y ) )
18	17	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( e = y -> ( ( # ` e ) = 2 <-> ( # ` y ) = 2 ) )
19	18	rspcv 3305	. . . . . . . . . . . . . . 15 |- ( y e. ran (iEdg ` G ) -> ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> ( # ` y ) = 2 ) )
20		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( # ` x ) = ( # ` y ) )
21	20	breq1d 4663	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( # ` x ) <_ 2 <-> ( # ` y ) <_ 2 ) )
22	21	elrab 3363	. . . . . . . . . . . . . . . 16 |- ( y e. { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( y e. ( ~P (Vtx ` G ) \ { (/) } ) /\ ( # ` y ) <_ 2 ) )
23		eldifi 3732	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. ( ~P (Vtx ` G ) \ { (/) } ) -> y e. ~P (Vtx ` G ) )
24	23	anim1i 592	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. ( ~P (Vtx ` G ) \ { (/) } ) /\ ( # ` y ) = 2 ) -> ( y e. ~P (Vtx ` G ) /\ ( # ` y ) = 2 ) )
25	20	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 |- ( x = y -> ( ( # ` x ) = 2 <-> ( # ` y ) = 2 ) )
26	25	elrab 3363	. . . . . . . . . . . . . . . . . . 19 |- ( y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } <-> ( y e. ~P (Vtx ` G ) /\ ( # ` y ) = 2 ) )
27	24, 26	sylibr 224	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. ( ~P (Vtx ` G ) \ { (/) } ) /\ ( # ` y ) = 2 ) -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
28	27	ex 450	. . . . . . . . . . . . . . . . 17 |- ( y e. ( ~P (Vtx ` G ) \ { (/) } ) -> ( ( # ` y ) = 2 -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
29	28	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( y e. ( ~P (Vtx ` G ) \ { (/) } ) /\ ( # ` y ) <_ 2 ) -> ( ( # ` y ) = 2 -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
30	22, 29	sylbi 207	. . . . . . . . . . . . . . 15 |- ( y e. { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ( ( # ` y ) = 2 -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
31	19, 30	syl9 77	. . . . . . . . . . . . . 14 |- ( y e. ran (iEdg ` G ) -> ( y e. { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) ) )
32	16, 31	syld 47	. . . . . . . . . . . . 13 |- ( y e. ran (iEdg ` G ) -> ( ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) ) )
33	32	com13 88	. . . . . . . . . . . 12 |- ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> ( ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ( y e. ran (iEdg ` G ) -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) ) )
34	33	imp 445	. . . . . . . . . . 11 |- ( ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 /\ ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) -> ( y e. ran (iEdg ` G ) -> y e. { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
35	34	ssrdv 3609	. . . . . . . . . 10 |- ( ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 /\ ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
36	35	ex 450	. . . . . . . . 9 |- ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> ( ran (iEdg ` G ) C_ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
37	14, 36	mpan9 486	. . . . . . . 8 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ A. e e. ran (iEdg ` G ) ( # ` e ) = 2 ) -> ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
38		f1ssr 6107	. . . . . . . 8 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ ran (iEdg ` G ) C_ { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
39	37, 38	syldan 487	. . . . . . 7 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ A. e e. ran (iEdg ` G ) ( # ` e ) = 2 ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
40	39	ex 450	. . . . . 6 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
41	11, 40	syl 17	. . . . 5 |- ( G e. USPGraph -> ( A. e e. ran (iEdg ` G ) ( # ` e ) = 2 -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
42	8, 41	sylbid 230	. . . 4 |- ( G e. USPGraph -> ( A. e e. (Edg ` G ) ( # ` e ) = 2 -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
43	42	imp 445	. . 3 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } )
44	9, 10	isusgrs 26051	. . . 4 |- ( G e. USPGraph -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
45	44	adantr 481	. . 3 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) -> ( G e. USGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ~P (Vtx ` G ) | ( # ` x ) = 2 } ) )
46	43, 45	mpbird 247	. 2 |- ( ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) -> G e. USGraph )
47	5, 46	impbii 199	1 |- ( G e. USGraph <-> ( G e. USPGraph /\ A. e e. (Edg ` G ) ( # ` e ) = 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   ran crn 5115   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USPGraph cuspgr 26043   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uspgr 26045  df-usgr 26046

This theorem is referenced by:  usgr1e  26137