Theorem subumgr 16515

Description: A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)

Assertion

Ref	Expression
subumgr	|- ( ( G e. UMGraph /\ S SubGraph G ) -> S e. UMGraph )

Proof of Theorem subumgr

Dummy variables x e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2238	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2238	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16501	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		umgruhgr 16354	. . . . . . . . . 10 |- ( G e. UMGraph -> G e. UHGraph )
8		subgruhgrfun 16509	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 283	. . . . . . . . 9 |- ( ( G e. UMGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 268	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. UMGraph ) -> Fun (iEdg ` S ) )
11	10	funfnd 5408	. . . . . . 7 |- ( ( S SubGraph G /\ G e. UMGraph ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
12	11	adantl 277	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
13		simplrl 541	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) /\ x e. dom (iEdg ` S ) ) -> S SubGraph G )
14		simplrr 542	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) /\ x e. dom (iEdg ` S ) ) -> G e. UMGraph )
15		simpr 110	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` S ) )
16	1, 3	subumgredg2en 16512	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. UMGraph /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
17	13, 14, 15, 16	syl3anc 1278	. . . . . . . 8 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
18	17	ralrimiva 2623	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
19		fnfvrnss 5868	. . . . . . 7 |- ( ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } ) -> ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | e ~~ 2o } )
20	12, 18, 19	syl2anc 415	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | e ~~ 2o } )
21		df-f 5381	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } <-> ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
22	12, 20, 21	sylanbrc 421	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } )
23		subgrv 16497	. . . . . . . 8 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
24	23	simpld 112	. . . . . . 7 |- ( S SubGraph G -> S e. _V )
25	1, 3	isumgren 16346	. . . . . . 7 |- ( S e. _V -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
26	24, 25	syl 14	. . . . . 6 |- ( S SubGraph G -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
27	26	ad2antrl 494	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
28	22, 27	mpbird 167	. . . 4 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UMGraph ) ) -> S e. UMGraph )
29	28	ex 115	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( ( S SubGraph G /\ G e. UMGraph ) -> S e. UMGraph ) )
30	6, 29	syl 14	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. UMGraph ) -> S e. UMGraph ) )
31	30	anabsi8 588	1 |- ( ( G e. UMGraph /\ S SubGraph G ) -> S e. UMGraph )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   dom cdm 4774   ran crn 4775   Fun wfun 5371   Fn wfn 5372   -->wf 5373   ` cfv 5377   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UHGraphcuhgr 16308  UMGraphcumgr 16333   SubGraph csubgr 16494

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-nul 4259  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-nul 3521  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-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  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-uhgrm 16310  df-upgren 16334  df-umgren 16335  df-subgr 16495

This theorem is used by:  umgrspan  16521