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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.
StepHypRef 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 )
61, 2, 3, 4, 5subgrprop2 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
) )
97, 8sylan 474 . . . . . . . . . 10  |-  ( ( G  e. USGraph  /\  S SubGraph  G )  ->  Fun  (iEdg `  S
) )
109ancoms 455 . . . . . . . . 9  |-  ( ( S SubGraph  G  /\  G  e. USGraph  )  ->  Fun  (iEdg `  S
) )
11 funfn 5611 . . . . . . . . 9  |-  ( Fun  (iEdg `  S )  <->  (iEdg `  S )  Fn  dom  (iEdg `  S ) )
1210, 11sylib 200 . . . . . . . 8  |-  ( ( S SubGraph  G  /\  G  e. USGraph  )  ->  (iEdg `  S
)  Fn  dom  (iEdg `  S ) )
1312adantl 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  )
1615adantl 468 . . . . . . . . . . . 12  |-  ( ( S SubGraph  G  /\  G  e. USGraph  )  ->  G  e. UMGraph  )
1716adantl 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  )
1817adantr 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 ) )
201, 3subumgredg2 39357 . . . . . . . . . 10  |-  ( ( S SubGraph  G  /\  G  e. UMGraph  /\  x  e.  dom  (iEdg `  S ) )  ->  ( (iEdg `  S ) `  x
)  e.  { e  e.  ~P (Vtx `  S )  |  (
# `  e )  =  2 } )
2114, 18, 19, 20syl3anc 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 } )
2221ralrimiva 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 } )
2413, 22, 23syl2anc 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 } ) )
2613, 24, 25sylanbrc 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 ) )
282, 4usgrfs 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 )
)
3130anim1i 572 . . . . . . . . . . . 12  |-  ( ( (iEdg `  G ) : dom  (iEdg `  G
) --> { y  e. 
~P (Vtx `  G
)  |  ( # `  y )  =  2 }  /\  Fun  `' (iEdg `  G ) )  ->  ( Fun  (iEdg `  G )  /\  Fun  `' (iEdg `  G )
) )
3229, 31sylbi 199 . . . . . . . . . . 11  |-  ( (iEdg `  G ) : dom  (iEdg `  G ) -1-1-> {
y  e.  ~P (Vtx `  G )  |  (
# `  y )  =  2 }  ->  ( Fun  (iEdg `  G
)  /\  Fun  `' (iEdg `  G ) ) )
3328, 32syl 17 . . . . . . . . . 10  |-  ( G  e. USGraph  ->  ( Fun  (iEdg `  G )  /\  Fun  `' (iEdg `  G )
) )
3433adantl 468 . . . . . . . . 9  |-  ( ( S SubGraph  G  /\  G  e. USGraph  )  ->  ( Fun  (iEdg `  G )  /\  Fun  `' (iEdg `  G )
) )
3527, 34anim12ci 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 ) ) )
3735, 36sylibr 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 )
)
3937, 38syl 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 )
) )
4126, 39, 40sylanbrc 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 ) )
431, 3isusgrs 39243 . . . . . . . . 9  |-  ( S  e.  _V  ->  ( S  e. USGraph  <->  (iEdg `  S ) : dom  (iEdg `  S
) -1-1-> { e  e.  ~P (Vtx `  S )  |  ( # `  e
)  =  2 } ) )
4443adantr 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 } ) )
4542, 44syl 17 . . . . . . 7  |-  ( S SubGraph  G  ->  ( S  e. USGraph  <->  (iEdg `  S ) : dom  (iEdg `  S ) -1-1-> {
e  e.  ~P (Vtx `  S )  |  (
# `  e )  =  2 } ) )
4645adantr 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 } ) )
4746adantl 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 } ) )
4841, 47mpbird 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  )
4948ex 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  ) )
506, 49syl 17 . 2  |-  ( S SubGraph  G  ->  ( ( S SubGraph  G  /\  G  e. USGraph  )  ->  S  e. USGraph  ) )
5150anabsi8 829 1  |-  ( ( G  e. USGraph  /\  S SubGraph  G )  ->  S  e. USGraph  )
Colors of variables: wff setvar class
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
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