Theorem ausgrusgrben 16392

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 5677	. . . . 5 |- ( _I |` E ) : E -1-1-onto-> E
2		dff1o5 5646	. . . . . 6 |- ( ( _I |` E ) : E -1-1-onto-> E <-> ( ( _I |` E ) : E -1-1-> E /\ ran ( _I |` E ) = E ) )
3		f1ss 5602	. . . . . . . . . 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 5116	. . . . . . . . . . . 12 |- dom ( _I |` E ) = E
5	4	eqcomi 2242	. . . . . . . . . . 11 |- E = dom ( _I |` E )
6		f1eq2 5592	. . . . . . . . . . 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 5379	. . . . . 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 5142	. . . . . . . . 9 |- ran ( _I |` E ) = E
16	15	sseq1i 3274	. . . . . . . 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 391	. . . . 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 5596	. . . . 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 5106	. . . . 5 |- ( E e. Y -> ( _I |` E ) e. _V )
24		opiedgfv 16249	. . . . 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 4981	. . . 4 |- ( ( V e. X /\ E e. Y ) -> dom (iEdg ` <. V , ( _I |` E ) >. ) = dom ( _I |` E ) )
27		opvtxfv 16246	. . . . . . 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 3693	. . . . 5 |- ( ( V e. X /\ E e. Y ) -> ~P (Vtx ` <. V , ( _I |` E ) >. ) = ~P V )
30	29	rabeqdv 2815	. . . 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 5629	. . 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 16391	. 2 |- ( ( V e. X /\ E e. Y ) -> ( V G E <-> E C_ { x e. ~P V | x ~~ 2o } ) )
35		opexg 4366	. . . 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 2238	. . . 4 |- (Vtx ` <. V , ( _I |` E ) >. ) = (Vtx ` <. V , ( _I |` E ) >. )
38		eqid 2238	. . . 4 |- (iEdg ` <. V , ( _I |` E ) >. ) = (iEdg ` <. V , ( _I |` E ) >. )
39	37, 38	isusgren 16382	. . 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 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   <.cop 3711   class class class wbr 4128   {copab 4189   _I cid 4431   dom cdm 4772   ran crn 4773   |` cres 4774   Fn wfn 5370   -->wf 5371   -1-1->wf1 5372   -1-1-onto->wf1o 5374   ` cfv 5375   2oc2o 6675   ~~ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  USGraphcusgr 16378

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-usgren 16380

This theorem is referenced by: (None)