Theorem subusgr 16270

Description: A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)

Assertion

Ref	Expression
subusgr	|- ( ( G e. USGraph /\ S SubGraph G ) -> S e. USGraph )

Proof of Theorem subusgr

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

Step	Hyp	Ref	Expression
1		eqid 2232	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2232	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2232	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2232	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2232	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16255	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		usgruhgr 16184	. . . . . . . . . . 11 |- ( G e. USGraph -> G e. UHGraph )
8		subgruhgrfun 16263	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 283	. . . . . . . . . 10 |- ( ( G e. USGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 268	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. USGraph ) -> Fun (iEdg ` S ) )
11	10	funfnd 5383	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. USGraph ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
12	11	adantl 277	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
13		simplrl 537	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) /\ x e. dom (iEdg ` S ) ) -> S SubGraph G )
14		usgrumgr 16179	. . . . . . . . . . . . 13 |- ( G e. USGraph -> G e. UMGraph )
15	14	adantl 277	. . . . . . . . . . . 12 |- ( ( S SubGraph G /\ G e. USGraph ) -> G e. UMGraph )
16	15	adantl 277	. . . . . . . . . . 11 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> G e. UMGraph )
17	16	adantr 276	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) /\ x e. dom (iEdg ` S ) ) -> G e. UMGraph )
18		simpr 110	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` S ) )
19	1, 3	subumgredg2en 16266	. . . . . . . . . 10 |- ( ( S SubGraph G /\ G e. UMGraph /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
20	13, 17, 18, 19	syl3anc 1274	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
21	20	ralrimiva 2615	. . . . . . . 8 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | e ~~ 2o } )
22		fnfvrnss 5837	. . . . . . . 8 |- ( ( (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 } )
23	12, 21, 22	syl2anc 411	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> ran (iEdg ` S ) C_ { e e. ~P (Vtx ` S ) | e ~~ 2o } )
24		df-f 5356	. . . . . . 7 |- ( (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 } ) )
25	12, 23, 24	sylanbrc 417	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } )
26		simp2 1025	. . . . . . . . 9 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> (iEdg ` S ) C_ (iEdg ` G ) )
27	2, 4	usgrfen 16155	. . . . . . . . . . 11 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | y ~~ 2o } )
28		df-f1 5357	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | y ~~ 2o } <-> ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | y ~~ 2o } /\ Fun `' (iEdg ` G ) ) )
29		ffun 5511	. . . . . . . . . . . . 13 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | y ~~ 2o } -> Fun (iEdg ` G ) )
30	29	anim1i 340	. . . . . . . . . . . 12 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | y ~~ 2o } /\ Fun `' (iEdg ` G ) ) -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
31	28, 30	sylbi 121	. . . . . . . . . . 11 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | y ~~ 2o } -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
32	27, 31	syl 14	. . . . . . . . . 10 |- ( G e. USGraph -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
33	32	adantl 277	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. USGraph ) -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
34	26, 33	anim12ci 339	. . . . . . . 8 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
35		df-3an 1007	. . . . . . . 8 |- ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) <-> ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
36	34, 35	sylibr 134	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
37		f1ssf1 5646	. . . . . . 7 |- ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) -> Fun `' (iEdg ` S ) )
38	36, 37	syl 14	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> Fun `' (iEdg ` S ) )
39		df-f1 5357	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } <-> ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | e ~~ 2o } /\ Fun `' (iEdg ` S ) ) )
40	25, 38, 39	sylanbrc 417	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } )
41		subgrv 16251	. . . . . . . . 9 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
42	41	simpld 112	. . . . . . . 8 |- ( S SubGraph G -> S e. _V )
43	1, 3	isusgren 16153	. . . . . . . 8 |- ( S e. _V -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
44	42, 43	syl 14	. . . . . . 7 |- ( S SubGraph G -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
45	44	adantr 276	. . . . . 6 |- ( ( S SubGraph G /\ G e. USGraph ) -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
46	45	adantl 277	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | e ~~ 2o } ) )
47	40, 46	mpbird 167	. . . 4 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. USGraph ) ) -> S e. USGraph )
48	47	ex 115	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( ( S SubGraph G /\ G e. USGraph ) -> S e. USGraph ) )
49	6, 48	syl 14	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. USGraph ) -> S e. USGraph ) )
50	49	anabsi8 584	1 |- ( ( G e. USGraph /\ S SubGraph G ) -> S e. USGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2203   A.wral 2520   {crab 2524   _Vcvv 2813   C_ wss 3211   ~Pcpw 3669   class class class wbr 4109   `'ccnv 4748   dom cdm 4749   ran crn 4750   Fun wfun 5346   Fn wfn 5347   -->wf 5348   -1-1->wf1 5349   ` cfv 5352   2oc2o 6641   ~~ cen 6973  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  UHGraphcuhgr 16062  UMGraphcumgr 16087  USGraphcusgr 16149   SubGraph csubgr 16248

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 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-en 6976  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-uhgrm 16064  df-upgren 16088  df-umgren 16089  df-uspgren 16150  df-usgren 16151  df-subgr 16249

This theorem is referenced by:  usgrspan  16276