Theorem ausgrusgrben 16289

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 5659	. . . . 5 |- ( _I |` E ) : E -1-1-onto-> E
2		dff1o5 5628	. . . . . 6 |- ( ( _I |` E ) : E -1-1-onto-> E <-> ( ( _I |` E ) : E -1-1-> E /\ ran ( _I |` E ) = E ) )
3		f1ss 5584	. . . . . . . . . 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 5098	. . . . . . . . . . . 12 |- dom ( _I |` E ) = E
5	4	eqcomi 2238	. . . . . . . . . . 11 |- E = dom ( _I |` E )
6		f1eq2 5574	. . . . . . . . . . 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 5361	. . . . . 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 5124	. . . . . . . . 9 |- ran ( _I |` E ) = E
16	15	sseq1i 3268	. . . . . . . 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 5578	. . . . 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 5088	. . . . 5 |- ( E e. Y -> ( _I |` E ) e. _V )
24		opiedgfv 16146	. . . . 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 4963	. . . 4 |- ( ( V e. X /\ E e. Y ) -> dom (iEdg ` <. V , ( _I |` E ) >. ) = dom ( _I |` E ) )
27		opvtxfv 16143	. . . . . . 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 3679	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> ~P (Vtx ` <. V , ( _I |` E ) >. ) = ~P V )
30	29	rabeqdv 2809	. . . 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 5611	. . 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 16288	. 2 |- ( ( V e. X /\ E e. Y ) -> ( V G E <-> E C_ { x e. ~P V | x ~~ 2o } ) )
35		opexg 4349	. . . 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 2234	. . . 4 |- (Vtx ` <. V , ( _I |` E ) >. ) = (Vtx ` <. V , ( _I |` E ) >. )
38		eqid 2234	. . . 4 |- (iEdg ` <. V , ( _I |` E ) >. ) = (iEdg ` <. V , ( _I |` E ) >. )
39	37, 38	isusgren 16279	. . 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 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   <.cop 3697   class class class wbr 4114   {copab 4175   _I cid 4414   dom cdm 4754   ran crn 4755   |` cres 4756   Fn wfn 5352   -->wf 5353   -1-1->wf1 5354   -1-1-onto->wf1o 5356   ` cfv 5357   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16133  iEdgciedg 16134  USGraphcusgr 16275

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-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-usgren 16277

This theorem is referenced by: (None)