Theorem ausgrumgrien 16182

Description: If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.)

Hypothesis

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

Assertion

Ref	Expression
ausgrumgrien	|- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ Fun (iEdg ` H ) ) -> H e. UMGraph )

Distinct variable group:   v, e, x, H

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

Proof of Theorem ausgrumgrien

Step	Hyp	Ref	Expression
1		vtxex 16030	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 16071	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 16031	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 5024	. . . . . . 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 2311	. . . . 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 16179	. . . . 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 3269	. . . . 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		funfn 5384	. . . . . . . . 9 |- ( Fun (iEdg ` H ) <-> (iEdg ` H ) Fn dom (iEdg ` H ) )
12	11	biimpi 120	. . . . . . . 8 |- ( Fun (iEdg ` H ) -> (iEdg ` H ) Fn dom (iEdg ` H ) )
13	12	3ad2ant3 1047	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ Fun (iEdg ` H ) ) -> (iEdg ` H ) Fn dom (iEdg ` H ) )
14		simp2 1025	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ Fun (iEdg ` H ) ) -> ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } )
15		df-f 5358	. . . . . . 7 |- ( (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } <-> ( (iEdg ` H ) Fn dom (iEdg ` H ) /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
16	13, 14, 15	sylanbrc 417	. . . . . 6 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ Fun (iEdg ` H ) ) -> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
17	16	3exp 1229	. . . . 5 |- ( H e. W -> ( ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( Fun (iEdg ` H ) -> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
18	10, 17	sylbid 150	. . . 4 |- ( H e. W -> ( (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( Fun (iEdg ` H ) -> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
19	9, 18	sylbid 150	. . 3 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) -> ( Fun (iEdg ` H ) -> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
20	19	3imp 1220	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ Fun (iEdg ` H ) ) -> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
21		eqid 2234	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
22		eqid 2234	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
23	21, 22	isumgren 16117	. . 3 |- ( H e. W -> ( H e. UMGraph <-> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
24	23	3ad2ant1 1045	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ Fun (iEdg ` H ) ) -> ( H e. UMGraph <-> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
25	20, 24	mpbird 167	1 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ Fun (iEdg ` H ) ) -> H e. UMGraph )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   C_ wss 3213   ~Pcpw 3671   class class class wbr 4111   {copab 4172   dom cdm 4751   ran crn 4752   Fun wfun 5348   Fn wfn 5349   -->wf 5350   ` cfv 5354   2oc2o 6643   ~~ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  Edgcedg 16069  UMGraphcumgr 16104

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 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-umgren 16106

This theorem is referenced by: (None)