Theorem subusgr 39361

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 2451	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2451	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2451	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2451	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2451	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 39346	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		usgruhgr 39270	. . . . . . . . . . 11 |- ( G e. USGraph -> G e. UHGraph )
8		subgruhgrfun 39354	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 474	. . . . . . . . . 10 |- ( ( G e. USGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 455	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. USGraph ) -> Fun (iEdg ` S ) )
11		funfn 5611	. . . . . . . . 9 |- ( Fun (iEdg ` S ) <-> (iEdg ` S ) Fn dom (iEdg ` S ) )
12	10, 11	sylib 200	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. USGraph ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
13	12	adantl 468	. . . . . . 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 ) )
14		simplrl 770	. . . . . . . . . 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 )
15		usgrumgr 39266	. . . . . . . . . . . . 13 |- ( G e. USGraph -> G e. UMGraph )
16	15	adantl 468	. . . . . . . . . . . 12 |- ( ( S SubGraph G /\ G e. USGraph ) -> G e. UMGraph )
17	16	adantl 468	. . . . . . . . . . 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 )
18	17	adantr 467	. . . . . . . . . 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 )
19		simpr 463	. . . . . . . . . 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 ) )
20	1, 3	subumgredg2 39357	. . . . . . . . . 10 |- ( ( S SubGraph G /\ G e. UMGraph /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } )
21	14, 18, 19, 20	syl3anc 1268	. . . . . . . . 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 ) = 2 } )
22	21	ralrimiva 2802	. . . . . . . 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 ) = 2 } )
23		fnfvrnss 6051	. . . . . . . 8 |- ( ( (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 } )
24	13, 22, 23	syl2anc 667	. . . . . . 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 ) = 2 } )
25		df-f 5586	. . . . . . 7 |- ( (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 } ) )
26	13, 24, 25	sylanbrc 670	. . . . . 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 ) = 2 } )
27		simp2 1009	. . . . . . . . 9 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> (iEdg ` S ) C_ (iEdg ` G ) )
28	2, 4	usgrfs 39244	. . . . . . . . . . 11 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } )
29		df-f1 5587	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } <-> ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } /\ Fun `' (iEdg ` G ) ) )
30		ffun 5731	. . . . . . . . . . . . 13 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } -> Fun (iEdg ` G ) )
31	30	anim1i 572	. . . . . . . . . . . 12 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) --> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } /\ Fun `' (iEdg ` G ) ) -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
32	29, 31	sylbi 199	. . . . . . . . . . 11 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { y e. ~P (Vtx ` G ) | ( # ` y ) = 2 } -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
33	28, 32	syl 17	. . . . . . . . . 10 |- ( G e. USGraph -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
34	33	adantl 468	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. USGraph ) -> ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) )
35	27, 34	anim12ci 571	. . . . . . . 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 ) ) )
36		df-3an 987	. . . . . . . 8 |- ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) <-> ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
37	35, 36	sylibr 216	. . . . . . 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 ) ) )
38		f1ssf1 39021	. . . . . . 7 |- ( ( Fun (iEdg ` G ) /\ Fun `' (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) ) -> Fun `' (iEdg ` S ) )
39	37, 38	syl 17	. . . . . 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 ) )
40		df-f1 5587	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } <-> ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } /\ Fun `' (iEdg ` S ) ) )
41	26, 39, 40	sylanbrc 670	. . . . 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 ) = 2 } )
42		subgrv 39342	. . . . . . . 8 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
43	1, 3	isusgrs 39243	. . . . . . . . 9 |- ( S e. _V -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
44	43	adantr 467	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
45	42, 44	syl 17	. . . . . . 7 |- ( S SubGraph G -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
46	45	adantr 467	. . . . . 6 |- ( ( S SubGraph G /\ G e. USGraph ) -> ( S e. USGraph <-> (iEdg ` S ) : dom (iEdg ` S ) -1-1-> { e e. ~P (Vtx ` S ) | ( # ` e ) = 2 } ) )
47	46	adantl 468	. . . . 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 ) = 2 } ) )
48	41, 47	mpbird 236	. . . 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 )
49	48	ex 436	. . 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 ) )
50	6, 49	syl 17	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. USGraph ) -> S e. USGraph ) )
51	50	anabsi8 829	1 |- ( ( G e. USGraph /\ S SubGraph G ) -> S e. USGraph )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   {crab 2741   _Vcvv 3045   C_ wss 3404   ~Pcpw 3951   class class class wbr 4402   `'ccnv 4833   dom cdm 4834   ran crn 4835   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   ` cfv 5582   2c2 10659   #chash 12515  Vtxcvtx 39101  iEdgciedg 39102   UHGraph cuhgr 39147   UMGraph cumgr 39173  Edgcedga 39210   USGraph cusgr 39236   SubGraph csubgr 39339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-uhgr 39149  df-upgr 39174  df-umgr 39175  df-edga 39211  df-uspgr 39237  df-usgr 39238  df-subgr 39340

This theorem is referenced by:  usgrspan  39367