Theorem ausgrumgrien 16152

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 16000	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 16041	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 16001	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 5021	. . . . . . 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 2309	. . . . 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 16149	. . . . 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 3266	. . . . 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 5381	. . . . . . . . 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 5355	. . . . . . 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 2232	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
22		eqid 2232	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
23	21, 22	isumgren 16087	. . 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 2203   {crab 2524   _Vcvv 2812   C_ wss 3210   ~Pcpw 3668   class class class wbr 4108   {copab 4169   dom cdm 4748   ran crn 4749   Fun wfun 5345   Fn wfn 5346   -->wf 5347   ` cfv 5351   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15994  iEdgciedg 15995  Edgcedg 16039  UMGraphcumgr 16074

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-umgren 16076

This theorem is referenced by: (None)