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Theorem umgrares 21364
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 21357 . . . 4  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 resss 5173 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5072 . . . . 5  |-  ( ( E  |`  A )  C_  E  ->  dom  ( E  |`  A )  C_  dom  E )
42, 3mp1i 12 . . . 4  |-  ( V UMGrph  E  ->  dom  ( E  |`  A )  C_  dom  E )
5 fssres 5613 . . . 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 644 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
7 resdmres 5364 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
87feq1i 5588 . . 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 190 . 2  |-  ( V UMGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
10 relumgra 21354 . . . 4  |-  Rel UMGrph
1110brrelexi 4921 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
1210brrelex2i 4922 . . . 4  |-  ( V UMGrph  E  ->  E  e.  _V )
13 resexg 5188 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1412, 13syl 16 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  A )  e.  _V )
15 isumgra 21355 . . 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 644 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
179, 16mpbird 225 1  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   class class class wbr 4215   dom cdm 4881    |` cres 4883   -->wf 5453   ` cfv 5457    <_ cle 9126   2c2 10054   #chash 11623   UMGrph cumg 21352
This theorem is referenced by:  eupares  21702  eupath2lem3  21706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-fun 5459  df-fn 5460  df-f 5461  df-umgra 21353
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