Theorem subupgr 26179

Description: A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)

Assertion

Ref	Expression
subupgr	|- ( ( G e. UPGraph /\ S SubGraph G ) -> S e. UPGraph )

Proof of Theorem subupgr

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		upgruhgr 25997	. . . . . . . . . 10 |- ( G e. UPGraph -> G e. UHGraph )
8		subgruhgrfun 26174	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 488	. . . . . . . . 9 |- ( ( G e. UPGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 469	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. UPGraph ) -> 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. UPGraph ) -> (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. UPGraph ) ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
14	7	anim2i 593	. . . . . . . . . . . . . 14 |- ( ( S SubGraph G /\ G e. UPGraph ) -> ( S SubGraph G /\ G e. UHGraph ) )
15	14	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( S SubGraph G /\ G e. UHGraph ) )
16	15	ancomd 467	. . . . . . . . . . . 12 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( G e. UHGraph /\ S SubGraph G ) )
17	16	anim1i 592	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( G e. UHGraph /\ S SubGraph G ) /\ x e. dom (iEdg ` S ) ) )
18	17	simplld 791	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> G e. UHGraph )
19		simpl 473	. . . . . . . . . . . 12 |- ( ( S SubGraph G /\ G e. UPGraph ) -> S SubGraph G )
20	19	adantl 482	. . . . . . . . . . 11 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> S SubGraph G )
21	20	adantr 481	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> S SubGraph G )
22		simpr 477	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` S ) )
23	1, 3, 18, 21, 22	subgruhgredgd 26176	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. ( ~P (Vtx ` S ) \ { (/) } ) )
24	4	uhgrfun 25961	. . . . . . . . . . . . . . . 16 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
25	7, 24	syl 17	. . . . . . . . . . . . . . 15 |- ( G e. UPGraph -> Fun (iEdg ` G ) )
26	25	ad2antll 765	. . . . . . . . . . . . . 14 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> Fun (iEdg ` G ) )
27	26	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> Fun (iEdg ` G ) )
28		simpll2 1101	. . . . . . . . . . . . 13 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> (iEdg ` S ) C_ (iEdg ` G ) )
29		funssfv 6209	. . . . . . . . . . . . 13 |- ( ( Fun (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` G ) ` x ) = ( (iEdg ` S ) ` x ) )
30	27, 28, 22, 29	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` G ) ` x ) = ( (iEdg ` S ) ` x ) )
31	30	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) = ( (iEdg ` G ) ` x ) )
32	31	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( # ` ( (iEdg ` S ) ` x ) ) = ( # ` ( (iEdg ` G ) ` x ) ) )
33		subgreldmiedg 26175	. . . . . . . . . . . . . . 15 |- ( ( S SubGraph G /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` G ) )
34	33	ex 450	. . . . . . . . . . . . . 14 |- ( S SubGraph G -> ( x e. dom (iEdg ` S ) -> x e. dom (iEdg ` G ) ) )
35	34	adantr 481	. . . . . . . . . . . . 13 |- ( ( S SubGraph G /\ G e. UPGraph ) -> ( x e. dom (iEdg ` S ) -> x e. dom (iEdg ` G ) ) )
36	35	adantl 482	. . . . . . . . . . . 12 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` S ) -> x e. dom (iEdg ` G ) ) )
37		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> G e. UPGraph )
38		funfn 5918	. . . . . . . . . . . . . . . . 17 |- ( Fun (iEdg ` G ) <-> (iEdg ` G ) Fn dom (iEdg ` G ) )
39	25, 38	sylib 208	. . . . . . . . . . . . . . . 16 |- ( G e. UPGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
40	39	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
41		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> x e. dom (iEdg ` G ) )
42	2, 4	upgrle 25985	. . . . . . . . . . . . . . 15 |- ( ( G e. UPGraph /\ (iEdg ` G ) Fn dom (iEdg ` G ) /\ x e. dom (iEdg ` G ) ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 )
43	37, 40, 41, 42	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 )
44	43	expcom 451	. . . . . . . . . . . . 13 |- ( G e. UPGraph -> ( x e. dom (iEdg ` G ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 ) )
45	44	ad2antll 765	. . . . . . . . . . . 12 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` G ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 ) )
46	36, 45	syld 47	. . . . . . . . . . 11 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` S ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 ) )
47	46	imp 445	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( # ` ( (iEdg ` G ) ` x ) ) <_ 2 )
48	32, 47	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( # ` ( (iEdg ` S ) ` x ) ) <_ 2 )
49		fveq2 6191	. . . . . . . . . . 11 |- ( e = ( (iEdg ` S ) ` x ) -> ( # ` e ) = ( # ` ( (iEdg ` S ) ` x ) ) )
50	49	breq1d 4663	. . . . . . . . . 10 |- ( e = ( (iEdg ` S ) ` x ) -> ( ( # ` e ) <_ 2 <-> ( # ` ( (iEdg ` S ) ` x ) ) <_ 2 ) )
51	50	elrab 3363	. . . . . . . . 9 |- ( ( (iEdg ` S ) ` x ) e. { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } <-> ( ( (iEdg ` S ) ` x ) e. ( ~P (Vtx ` S ) \ { (/) } ) /\ ( # ` ( (iEdg ` S ) ` x ) ) <_ 2 ) )
52	23, 48, 51	sylanbrc 698	. . . . . . . 8 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } )
53	52	ralrimiva 2966	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } )
54		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 } )
55	13, 53, 54	syl2anc 693	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ran (iEdg ` S ) C_ { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } )
56		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 } ) )
57	13, 55, 56	sylanbrc 698	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } )
58		subgrv 26162	. . . . . . . 8 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
59	1, 3	isupgr 25979	. . . . . . . . 9 |- ( S e. _V -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } ) )
60	59	adantr 481	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } ) )
61	58, 60	syl 17	. . . . . . 7 |- ( S SubGraph G -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } ) )
62	61	adantr 481	. . . . . 6 |- ( ( S SubGraph G /\ G e. UPGraph ) -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } ) )
63	62	adantl 482	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ( ~P (Vtx ` S ) \ { (/) } ) | ( # ` e ) <_ 2 } ) )
64	57, 63	mpbird 247	. . . 4 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> S e. UPGraph )
65	64	ex 450	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( ( S SubGraph G /\ G e. UPGraph ) -> S e. UPGraph ) )
66	6, 65	syl 17	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. UPGraph ) -> S e. UPGraph ) )
67	66	anabsi8 861	1 |- ( ( G e. UPGraph /\ S SubGraph G ) -> S e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UPGraph cupgr 25975   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-subgr 26160

This theorem is referenced by:  upgrspan  26185