Theorem subusgr 16430

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 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 16415	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		usgruhgr 16344	. . . . . . . . . . 11 |- ( G e. USGraph -> G e. UHGraph )
8		subgruhgrfun 16423	. . . . . . . . . . 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 5403	. . . . . . . 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 541	. . . . . . . . . 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 16339	. . . . . . . . . . . . 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 16426	. . . . . . . . . 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 1278	. . . . . . . . 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 2623	. . . . . . . 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 5859	. . . . . . . 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 415	. . . . . . 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 5376	. . . . . . 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 421	. . . . . 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 1029	. . . . . . . . 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 16315	. . . . . . . . . . 11 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | y ~~ 2o } )
28		df-f1 5377	. . . . . . . . . . . 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 5531	. . . . . . . . . . . . 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 1011	. . . . . . . 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 5666	. . . . . . 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 5377	. . . . . 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 421	. . . . 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 16411	. . . . . . . . 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 16313	. . . . . . . 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 588	1 |- ( ( G e. USGraph /\ S SubGraph G ) -> S e. USGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   `'ccnv 4768   dom cdm 4769   ran crn 4770   Fun wfun 5366   Fn wfn 5367   -->wf 5368   -1-1->wf1 5369   ` cfv 5372   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UHGraphcuhgr 16222  UMGraphcumgr 16247  USGraphcusgr 16309   SubGraph csubgr 16408

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-upgren 16248  df-umgren 16249  df-uspgren 16310  df-usgren 16311  df-subgr 16409

This theorem is referenced by:  usgrspan  16436