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Theorem 0grsubgr 39514
Description: The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
Assertion
Ref Expression
0grsubgr  |-  ( G  e.  W  ->  (/) SubGraph  G )

Proof of Theorem 0grsubgr
StepHypRef Expression
1 0ss 3766 . . 3  |-  (/)  C_  (Vtx `  G )
2 dm0 5054 . . . . 5  |-  dom  (/)  =  (/)
32reseq2i 5108 . . . 4  |-  ( (iEdg `  G )  |`  dom  (/) )  =  ( (iEdg `  G
)  |`  (/) )
4 res0 5115 . . . 4  |-  ( (iEdg `  G )  |`  (/) )  =  (/)
53, 4eqtr2i 2494 . . 3  |-  (/)  =  ( (iEdg `  G )  |` 
dom  (/) )
6 0ss 3766 . . 3  |-  (/)  C_  ~P (/)
71, 5, 63pm3.2i 1208 . 2  |-  ( (/)  C_  (Vtx `  G )  /\  (/)  =  ( (iEdg `  G )  |`  dom  (/) )  /\  (/)  C_  ~P (/) )
8 0ex 4528 . . 3  |-  (/)  e.  _V
9 vtxval0 39292 . . . . 5  |-  (Vtx `  (/) )  =  (/)
109eqcomi 2480 . . . 4  |-  (/)  =  (Vtx
`  (/) )
11 eqid 2471 . . . 4  |-  (Vtx `  G )  =  (Vtx
`  G )
12 iedgval0 39293 . . . . 5  |-  (iEdg `  (/) )  =  (/)
1312eqcomi 2480 . . . 4  |-  (/)  =  (iEdg `  (/) )
14 eqid 2471 . . . 4  |-  (iEdg `  G )  =  (iEdg `  G )
15 edgaval 39373 . . . . . 6  |-  ( (/)  e.  _V  ->  (Edg `  (/) )  =  ran  (iEdg `  (/) ) )
168, 15ax-mp 5 . . . . 5  |-  (Edg `  (/) )  =  ran  (iEdg `  (/) )
1712rneqi 5067 . . . . 5  |-  ran  (iEdg `  (/) )  =  ran  (/)
18 rn0 5092 . . . . 5  |-  ran  (/)  =  (/)
1916, 17, 183eqtrri 2498 . . . 4  |-  (/)  =  (Edg
`  (/) )
2010, 11, 13, 14, 19issubgr 39507 . . 3  |-  ( ( G  e.  W  /\  (/) 
e.  _V )  ->  ( (/) SubGraph  G  <-> 
( (/)  C_  (Vtx `  G
)  /\  (/)  =  ( (iEdg `  G )  |` 
dom  (/) )  /\  (/)  C_  ~P (/) ) ) )
218, 20mpan2 685 . 2  |-  ( G  e.  W  ->  ( (/) SubGraph  G  <-> 
( (/)  C_  (Vtx `  G
)  /\  (/)  =  ( (iEdg `  G )  |` 
dom  (/) )  /\  (/)  C_  ~P (/) ) ) )
227, 21mpbiri 241 1  |-  ( G  e.  W  ->  (/) SubGraph  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   class class class wbr 4395   dom cdm 4839   ran crn 4840    |` cres 4841   ` cfv 5589  Vtxcvtx 39251  iEdgciedg 39252  Edgcedga 39371   SubGraph csubgr 39503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-slot 15203  df-base 15204  df-edgf 39246  df-vtx 39253  df-iedg 39254  df-edga 39372  df-subgr 39504
This theorem is referenced by: (None)
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