Theorem ausgrusgrben 15987

Description: The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)

Hypothesis

Ref	Expression
ausgr.1	|- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }

Assertion

Ref	Expression
ausgrusgrben	|- ( ( V e. X /\ E e. Y ) -> ( V G E <-> <. V , ( _I |` E ) >. e. USGraph ) )

Distinct variable groups:   v, e, x, E   e, V, v, x   x, X   x, Y

Allowed substitution hints:   G( x, v, e)   X( v, e)   Y( v, e)

Proof of Theorem ausgrusgrben

Step	Hyp	Ref	Expression
1		f1oi 5616	. . . . 5 |- ( _I |` E ) : E -1-1-onto-> E
2		dff1o5 5586	. . . . . 6 |- ( ( _I |` E ) : E -1-1-onto-> E <-> ( ( _I |` E ) : E -1-1-> E /\ ran ( _I |` E ) = E ) )
3		f1ss 5542	. . . . . . . . . 10 |- ( ( ( _I |` E ) : E -1-1-> E /\ E C_ { x e. ~P V | x ~~ 2o } ) -> ( _I |` E ) : E -1-1-> { x e. ~P V | x ~~ 2o } )
4		dmresi 5063	. . . . . . . . . . . 12 |- dom ( _I |` E ) = E
5	4	eqcomi 2233	. . . . . . . . . . 11 |- E = dom ( _I |` E )
6		f1eq2 5532	. . . . . . . . . . 11 |- ( E = dom ( _I |` E ) -> ( ( _I |` E ) : E -1-1-> { x e. ~P V | x ~~ 2o } <-> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) )
7	5, 6	ax-mp 5	. . . . . . . . . 10 |- ( ( _I |` E ) : E -1-1-> { x e. ~P V | x ~~ 2o } <-> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } )
8	3, 7	sylib 122	. . . . . . . . 9 |- ( ( ( _I |` E ) : E -1-1-> E /\ E C_ { x e. ~P V | x ~~ 2o } ) -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } )
9	8	ex 115	. . . . . . . 8 |- ( ( _I |` E ) : E -1-1-> E -> ( E C_ { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) )
10	9	a1d 22	. . . . . . 7 |- ( ( _I |` E ) : E -1-1-> E -> ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) ) )
11	10	adantr 276	. . . . . 6 |- ( ( ( _I |` E ) : E -1-1-> E /\ ran ( _I |` E ) = E ) -> ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) ) )
12	2, 11	sylbi 121	. . . . 5 |- ( ( _I |` E ) : E -1-1-onto-> E -> ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) ) )
13	1, 12	ax-mp 5	. . . 4 |- ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) )
14		df-f 5325	. . . . . 6 |- ( ( _I |` E ) : dom ( _I |` E ) --> { x e. ~P V | x ~~ 2o } <-> ( ( _I |` E ) Fn dom ( _I |` E ) /\ ran ( _I |` E ) C_ { x e. ~P V | x ~~ 2o } ) )
15		rnresi 5088	. . . . . . . . 9 |- ran ( _I |` E ) = E
16	15	sseq1i 3250	. . . . . . . 8 |- ( ran ( _I |` E ) C_ { x e. ~P V | x ~~ 2o } <-> E C_ { x e. ~P V | x ~~ 2o } )
17	16	biimpi 120	. . . . . . 7 |- ( ran ( _I |` E ) C_ { x e. ~P V | x ~~ 2o } -> E C_ { x e. ~P V | x ~~ 2o } )
18	17	a1d 22	. . . . . 6 |- ( ran ( _I |` E ) C_ { x e. ~P V | x ~~ 2o } -> ( ( V e. X /\ E e. Y ) -> E C_ { x e. ~P V | x ~~ 2o } ) )
19	14, 18	simplbiim 387	. . . . 5 |- ( ( _I |` E ) : dom ( _I |` E ) --> { x e. ~P V | x ~~ 2o } -> ( ( V e. X /\ E e. Y ) -> E C_ { x e. ~P V | x ~~ 2o } ) )
20		f1f 5536	. . . . 5 |- ( ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } -> ( _I |` E ) : dom ( _I |` E ) --> { x e. ~P V | x ~~ 2o } )
21	19, 20	syl11 31	. . . 4 |- ( ( V e. X /\ E e. Y ) -> ( ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } -> E C_ { x e. ~P V | x ~~ 2o } ) )
22	13, 21	impbid 129	. . 3 |- ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } <-> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) )
23		resiexg 5053	. . . . 5 |- ( E e. Y -> ( _I |` E ) e. _V )
24		opiedgfv 15847	. . . . 5 |- ( ( V e. X /\ ( _I |` E ) e. _V ) -> (iEdg ` <. V , ( _I |` E ) >. ) = ( _I |` E ) )
25	23, 24	sylan2 286	. . . 4 |- ( ( V e. X /\ E e. Y ) -> (iEdg ` <. V , ( _I |` E ) >. ) = ( _I |` E ) )
26	25	dmeqd 4928	. . . 4 |- ( ( V e. X /\ E e. Y ) -> dom (iEdg ` <. V , ( _I |` E ) >. ) = dom ( _I |` E ) )
27		opvtxfv 15844	. . . . . . 7 |- ( ( V e. X /\ ( _I |` E ) e. _V ) -> (Vtx ` <. V , ( _I |` E ) >. ) = V )
28	23, 27	sylan2 286	. . . . . 6 |- ( ( V e. X /\ E e. Y ) -> (Vtx ` <. V , ( _I |` E ) >. ) = V )
29	28	pweqd 3654	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> ~P (Vtx ` <. V , ( _I |` E ) >. ) = ~P V )
30	29	rabeqdv 2793	. . . 4 |- ( ( V e. X /\ E e. Y ) -> { x e. ~P (Vtx ` <. V , ( _I |` E ) >. ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
31	25, 26, 30	f1eq123d 5569	. . 3 |- ( ( V e. X /\ E e. Y ) -> ( (iEdg ` <. V , ( _I |` E ) >. ) : dom (iEdg ` <. V , ( _I |` E ) >. ) -1-1-> { x e. ~P (Vtx ` <. V , ( _I |` E ) >. ) | x ~~ 2o } <-> ( _I |` E ) : dom ( _I |` E ) -1-1-> { x e. ~P V | x ~~ 2o } ) )
32	22, 31	bitr4d 191	. 2 |- ( ( V e. X /\ E e. Y ) -> ( E C_ { x e. ~P V | x ~~ 2o } <-> (iEdg ` <. V , ( _I |` E ) >. ) : dom (iEdg ` <. V , ( _I |` E ) >. ) -1-1-> { x e. ~P (Vtx ` <. V , ( _I |` E ) >. ) | x ~~ 2o } ) )
33		ausgr.1	. . 3 |- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
34	33	isausgren 15986	. 2 |- ( ( V e. X /\ E e. Y ) -> ( V G E <-> E C_ { x e. ~P V | x ~~ 2o } ) )
35		opexg 4315	. . . 4 |- ( ( V e. X /\ ( _I |` E ) e. _V ) -> <. V , ( _I |` E ) >. e. _V )
36	23, 35	sylan2 286	. . 3 |- ( ( V e. X /\ E e. Y ) -> <. V , ( _I |` E ) >. e. _V )
37		eqid 2229	. . . 4 |- (Vtx ` <. V , ( _I |` E ) >. ) = (Vtx ` <. V , ( _I |` E ) >. )
38		eqid 2229	. . . 4 |- (iEdg ` <. V , ( _I |` E ) >. ) = (iEdg ` <. V , ( _I |` E ) >. )
39	37, 38	isusgren 15977	. . 3 |- ( <. V , ( _I |` E ) >. e. _V -> ( <. V , ( _I |` E ) >. e. USGraph <-> (iEdg ` <. V , ( _I |` E ) >. ) : dom (iEdg ` <. V , ( _I |` E ) >. ) -1-1-> { x e. ~P (Vtx ` <. V , ( _I |` E ) >. ) | x ~~ 2o } ) )
40	36, 39	syl 14	. 2 |- ( ( V e. X /\ E e. Y ) -> ( <. V , ( _I |` E ) >. e. USGraph <-> (iEdg ` <. V , ( _I |` E ) >. ) : dom (iEdg ` <. V , ( _I |` E ) >. ) -1-1-> { x e. ~P (Vtx ` <. V , ( _I |` E ) >. ) | x ~~ 2o } ) )
41	32, 34, 40	3bitr4d 220	1 |- ( ( V e. X /\ E e. Y ) -> ( V G E <-> <. V , ( _I |` E ) >. e. USGraph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   <.cop 3669   class class class wbr 4083   {copab 4144   _I cid 4380   dom cdm 4720   ran crn 4721   |` cres 4722   Fn wfn 5316   -->wf 5317   -1-1->wf1 5318   -1-1-onto->wf1o 5320   ` cfv 5321   2oc2o 6567   ~~ cen 6898  Vtxcvtx 15834  iEdgciedg 15835  USGraphcusgr 15973

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-sub 8335  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-dec 9595  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-usgren 15975

This theorem is referenced by: (None)