Theorem subuhgr 26178

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

Assertion

Ref	Expression
subuhgr	|- ( ( G e. UHGraph /\ S SubGraph G ) -> S e. UHGraph )

Proof of Theorem subuhgr

Dummy variable x is 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		subgruhgrfun 26174	. . . . . . . . 9 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
8	7	ancoms 469	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. UHGraph ) -> Fun (iEdg ` S ) )
9	8	adantl 482	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> Fun (iEdg ` S ) )
10		funfn 5918	. . . . . . 7 |- ( Fun (iEdg ` S ) <-> (iEdg ` S ) Fn dom (iEdg ` S ) )
11	9, 10	sylib 208	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
12		simplrr 801	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) /\ x e. dom (iEdg ` S ) ) -> G e. UHGraph )
13		simplrl 800	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) /\ x e. dom (iEdg ` S ) ) -> S SubGraph G )
14		simpr 477	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` S ) )
15	1, 3, 12, 13, 14	subgruhgredgd 26176	. . . . . . . 8 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. ( ~P (Vtx ` S ) \ { (/) } ) )
16	15	ralrimiva 2966	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. ( ~P (Vtx ` S ) \ { (/) } ) )
17		fnfvrnss 6390	. . . . . . 7 |- ( ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. ( ~P (Vtx ` S ) \ { (/) } ) ) -> ran (iEdg ` S ) C_ ( ~P (Vtx ` S ) \ { (/) } ) )
18	11, 16, 17	syl2anc 693	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> ran (iEdg ` S ) C_ ( ~P (Vtx ` S ) \ { (/) } ) )
19		df-f 5892	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) <-> ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ ran (iEdg ` S ) C_ ( ~P (Vtx ` S ) \ { (/) } ) ) )
20	11, 18, 19	sylanbrc 698	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) )
21		subgrv 26162	. . . . . . . 8 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
22	1, 3	isuhgr 25955	. . . . . . . . 9 |- ( S e. _V -> ( S e. UHGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) ) )
23	22	adantr 481	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> ( S e. UHGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) ) )
24	21, 23	syl 17	. . . . . . 7 |- ( S SubGraph G -> ( S e. UHGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) ) )
25	24	adantr 481	. . . . . 6 |- ( ( S SubGraph G /\ G e. UHGraph ) -> ( S e. UHGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) ) )
26	25	adantl 482	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> ( S e. UHGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> ( ~P (Vtx ` S ) \ { (/) } ) ) )
27	20, 26	mpbird 247	. . . 4 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UHGraph ) ) -> S e. UHGraph )
28	27	ex 450	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( ( S SubGraph G /\ G e. UHGraph ) -> S e. UHGraph ) )
29	6, 28	syl 17	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. UHGraph ) -> S e. UHGraph ) )
30	29	anabsi8 861	1 |- ( ( G e. UHGraph /\ S SubGraph G ) -> S e. UHGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _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  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   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-subgr 26160

This theorem is referenced by:  uhgrspan  26184