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Theorem umgrares 21316
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 21309 . . . 4  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 resss 5133 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5032 . . . . 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 5573 . . . 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 643 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
7 resdmres 5324 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
87feq1i 5548 . . 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 189 . 2  |-  ( V UMGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
10 relumgra 21306 . . . 4  |-  Rel UMGrph
1110brrelexi 4881 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
1210brrelex2i 4882 . . . 4  |-  ( V UMGrph  E  ->  E  e.  _V )
13 resexg 5148 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1412, 13syl 16 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  A )  e.  _V )
15 isumgra 21307 . . 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 643 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
179, 16mpbird 224 1  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   {crab 2674   _Vcvv 2920    \ cdif 3281    C_ wss 3284   (/)c0 3592   ~Pcpw 3763   {csn 3778   class class class wbr 4176   dom cdm 4841    |` cres 4843   -->wf 5413   ` cfv 5417    <_ cle 9081   2c2 10009   #chash 11577   UMGrph cumg 21304
This theorem is referenced by:  eupares  21654  eupath2lem3  21658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-fun 5419  df-fn 5420  df-f 5421  df-umgra 21305
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