Theorem subumgr 16131

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 2231	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2231	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2231	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16117	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		umgruhgr 15970	. . . . . . . . . 10 |- ( G e. UMGraph -> G e. UHGraph )
8		subgruhgrfun 16125	. . . . . . . . . 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 5357	. . . . . . 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 537	. . . . . . . . 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 538	. . . . . . . . 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 16128	. . . . . . . . 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 1273	. . . . . . . 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 2605	. . . . . . 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 5807	. . . . . . 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 411	. . . . . 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 5330	. . . . . 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 417	. . . . 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 16113	. . . . . . . 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 15962	. . . . . . 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 490	. . . . 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 584	1 |- ( ( G e. UMGraph /\ S SubGraph G ) -> S e. UMGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   ran crn 4726   Fun wfun 5320   Fn wfn 5321   -->wf 5322   ` cfv 5326   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15869  iEdgciedg 15870  Edgcedg 15914  UHGraphcuhgr 15924  UMGraphcumgr 15949   SubGraph csubgr 16110

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872  df-edg 15915  df-uhgrm 15926  df-upgren 15950  df-umgren 15951  df-subgr 16111

This theorem is referenced by:  umgrspan  16137