Theorem ausgrusgrien 15969

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 15819	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 15860	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 15820	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 4989	. . . . . . 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 2306	. . . . 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 15965	. . . . 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 411	. . . 4 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) <-> (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
10	2	sseq1d 3253	. . . . 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 2296	. . . . . . . . . 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 4923	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> dom f = dom (iEdg ` H ) )
16		rneq 4951	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> ran f = ran (iEdg ` H ) )
17	14, 15, 16	f1eq123d 5564	. . . . . . . . . 10 |- ( f = (iEdg ` H ) -> ( f : dom f -1-1-> ran f <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
18	17	elabg 2949	. . . . . . . . 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 1044	. . . . . . 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 1022	. . . . . . 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 5538	. . . . . . 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 411	. . . . . 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 1226	. . . . 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 1217	. 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 2229	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
29		eqid 2229	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
30	28, 29	isusgren 15956	. . 3 |- ( H e. W -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
31	30	3ad2ant1 1042	. 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 )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   {copab 4144   dom cdm 4719   ran crn 4720   -1-1->wf1 5315   ` cfv 5318   2oc2o 6556   ~~ cen 6885  Vtxcvtx 15813  iEdgciedg 15814  Edgcedg 15858  USGraphcusgr 15952

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

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 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-sub 8319  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-dec 9579  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-usgren 15954

This theorem is referenced by:  usgrausgrben  15970