Theorem ausgrumgrien 16616

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 16464	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 16505	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 16465	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 5049	. . . . . . 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 16613	. . . . 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		funfn 5410	. . . . . . . . 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 1051	. . . . . . 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 1029	. . . . . . 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 5384	. . . . . . 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 421	. . . . . 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 1233	. . . . 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 1224	. 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 2238	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
22		eqid 2238	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
23	21, 22	isumgren 16551	. . 3 |- ( H e. W -> ( H e. UMGraph <-> (iEdg ` H ) : dom (iEdg ` H ) --> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
24	23	3ad2ant1 1049	. 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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   class class class wbr 4131   {copab 4192   dom cdm 4776   ran crn 4777   Fun wfun 5374   Fn wfn 5375   -->wf 5376   ` cfv 5380   2oc2o 6685   ~~ cen 7024  Vtxcvtx 16458  iEdgciedg 16459  Edgcedg 16503  UMGraphcumgr 16538

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sub 8506  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-dec 9789  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-edg 16504  df-umgren 16540

This theorem is used by: (None)