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Theorem umgrares 24280
Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
umgrares  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )

Proof of Theorem umgrares
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 umgraf2 24273 . . . 4  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 resss 5058 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 4957 . . . . 5  |-  ( ( E  |`  A )  C_  E  ->  dom  ( E  |`  A )  C_  dom  E )
42, 3mp1i 11 . . . 4  |-  ( V UMGrph  E  ->  dom  ( E  |`  A )  C_  dom  E )
5 fssres 5488 . . . 4  |-  ( ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }  /\  dom  ( E  |`  A )  C_  dom  E )  ->  ( E  |` 
dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
61, 4, 5syl2anc 642 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
7 resdmres 5243 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
87feq1i 5463 . . 3  |-  ( ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
96, 8sylib 188 . 2  |-  ( V UMGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
10 relumgra 24270 . . . 4  |-  Rel UMGrph
1110brrelexi 4808 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
1210brrelex2i 4809 . . . 4  |-  ( V UMGrph  E  ->  E  e.  _V )
13 resexg 5073 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1412, 13syl 15 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  A )  e.  _V )
15 isumgra 24271 . . 3  |-  ( ( V  e.  _V  /\  ( E  |`  A )  e.  _V )  -> 
( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
1611, 14, 15syl2anc 642 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
179, 16mpbird 223 1  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1710   {crab 2623   _Vcvv 2864    \ cdif 3225    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   {csn 3716   class class class wbr 4102   dom cdm 4768    |` cres 4770   -->wf 5330   ` cfv 5334    <_ cle 8955   2c2 9882   #chash 11427   UMGrph cumg 24264
This theorem is referenced by:  eupares  24303  eupath2lem3  24307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-fun 5336  df-fn 5337  df-f 5338  df-umgra 24267
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