Theorem ausgrusgrien 16412

Description: The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)

Hypotheses

Ref	Expression
ausgr.1	|- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
ausgrusgri.1	|- O = { f | f : dom f -1-1-> ran f }

Assertion

Ref	Expression
ausgrusgrien	|- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> H e. USGraph )

Distinct variable groups:   v, e, x, H   f, H   x, W

Allowed substitution hints:   G( x, v, e, f)   O( x, v, e, f)   W( v, e, f)

Proof of Theorem ausgrusgrien

Step	Hyp	Ref	Expression
1		vtxex 16259	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 16300	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 16260	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 5047	. . . . . . 7 |- ( (iEdg ` H ) e. _V -> ran (iEdg ` H ) e. _V )
5	3, 4	syl 14	. . . . . 6 |- ( H e. W -> ran (iEdg ` H ) e. _V )
6	2, 5	eqeltrd 2315	. . . . 5 |- ( H e. W -> (Edg ` H ) e. _V )
7		ausgr.1	. . . . . 6 |- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
8	7	isausgren 16408	. . . . 5 |- ( ( (Vtx ` H ) e. _V /\ (Edg ` H ) e. _V ) -> ( (Vtx ` H ) G (Edg ` H ) <-> (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
9	1, 6, 8	syl2anc 415	. . . 4 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) <-> (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
10	2	sseq1d 3277	. . . . 5 |- ( H e. W -> ( (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } <-> ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
11		ausgrusgri.1	. . . . . . . . . . 11 |- O = { f | f : dom f -1-1-> ran f }
12	11	eleq2i 2305	. . . . . . . . . 10 |- ( (iEdg ` H ) e. O <-> (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } )
13	12	biimpi 120	. . . . . . . . 9 |- ( (iEdg ` H ) e. O -> (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } )
14		id 19	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> f = (iEdg ` H ) )
15		dmeq 4981	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> dom f = dom (iEdg ` H ) )
16		rneq 5009	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> ran f = ran (iEdg ` H ) )
17	14, 15, 16	f1eq123d 5631	. . . . . . . . . 10 |- ( f = (iEdg ` H ) -> ( f : dom f -1-1-> ran f <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
18	17	elabg 2972	. . . . . . . . 9 |- ( (iEdg ` H ) e. O -> ( (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
19	13, 18	mpbid 147	. . . . . . . 8 |- ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) )
20	19	3ad2ant3 1051	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) )
21		simp2 1029	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } )
22		f1ssr 5605	. . . . . . 7 |- ( ( (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
23	20, 21, 22	syl2anc 415	. . . . . 6 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
24	23	3exp 1233	. . . . 5 |- ( H e. W -> ( ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
25	10, 24	sylbid 150	. . . 4 |- ( H e. W -> ( (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
26	9, 25	sylbid 150	. . 3 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
27	26	3imp 1224	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
28		eqid 2238	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
29		eqid 2238	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
30	28, 29	isusgren 16399	. . 3 |- ( H e. W -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
31	30	3ad2ant1 1049	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
32	27, 31	mpbird 167	1 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> H e. USGraph )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   {copab 4191   dom cdm 4774   ran crn 4775   -1-1->wf1 5374   ` cfv 5377   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  USGraphcusgr 16395

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-usgren 16397

This theorem is used by:  usgrausgrben  16413