Theorem subumgr 26180

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 2622	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2622	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 26166	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		umgruhgr 25999	. . . . . . . . . 10 |- ( G e. UMGraph -> G e. UHGraph )
8		subgruhgrfun 26174	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 488	. . . . . . . . 9 |- ( ( G e. UMGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 469	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. UMGraph ) -> Fun (iEdg ` S ) )
11		funfn 5918	. . . . . . . 8 |- ( Fun (iEdg ` S ) <-> (iEdg ` S ) Fn dom (iEdg ` S ) )
12	10, 11	sylib 208	. . . . . . 7 |- ( ( S SubGraph G /\ G e. UMGraph ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
13	12	adantl 482	. . . . . 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 ) )
14		simplrl 800	. . . . . . . . 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 )
15		simplrr 801	. . . . . . . . 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 )
16		simpr 477	. . . . . . . . 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 ) )
17	1, 3	subumgredg2 26177	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. UMGraph /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } )
18	14, 15, 16, 17	syl3anc 1326	. . . . . . . 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 ) = 2 } )
19	18	ralrimiva 2966	. . . . . . 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 ) = 2 } )
20		fnfvrnss 6390	. . . . . . 7 |- ( ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) -> ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } )
21	13, 19, 20	syl2anc 693	. . . . . 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 ) = 2 } )
22		df-f 5892	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } <-> ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
23	13, 21, 22	sylanbrc 698	. . . . 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 ) = 2 } )
24		subgrv 26162	. . . . . . 7 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
25	1, 3	isumgrs 25991	. . . . . . . 8 |- ( S e. _V -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
26	25	adantr 481	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
27	24, 26	syl 17	. . . . . 6 |- ( S SubGraph G -> ( S e. UMGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
28	27	ad2antrl 764	. . . . 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 ) = 2 } ) )
29	23, 28	mpbird 247	. . . 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 )
30	29	ex 450	. . 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 ) )
31	6, 30	syl 17	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. UMGraph ) -> S e. UMGraph ) )
32	31	anabsi8 861	1 |- ( ( G e. UMGraph /\ S SubGraph G ) -> S e. UMGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UMGraph cumgr 25976   SubGraph csubgr 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-subgr 26160

This theorem is referenced by:  umgrspan  26186