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Theorem umgrares 25047
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 25040 . . . 4  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 resss 5146 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5052 . . . . 5  |-  ( ( E  |`  A )  C_  E  ->  dom  ( E  |`  A )  C_  dom  E )
42, 3mp1i 13 . . . 4  |-  ( V UMGrph  E  ->  dom  ( E  |`  A )  C_  dom  E )
51, 4fssresd 5766 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
6 resdmres 5344 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
76feq1i 5737 . . 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 }
)
85, 7sylib 200 . 2  |-  ( V UMGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
9 relumgra 25037 . . . 4  |-  Rel UMGrph
109brrelexi 4893 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
119brrelex2i 4894 . . . 4  |-  ( V UMGrph  E  ->  E  e.  _V )
12 resexg 5165 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1311, 12syl 17 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  A )  e.  _V )
14 isumgra 25038 . . 3  |-  ( ( V  e.  _V  /\  ( E  |`  A )  e.  _V )  -> 
( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
1510, 13, 14syl2anc 666 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
168, 15mpbird 236 1  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    e. wcel 1869   {crab 2780   _Vcvv 3082    \ cdif 3435    C_ wss 3438   (/)c0 3763   ~Pcpw 3981   {csn 3998   class class class wbr 4422   dom cdm 4852    |` cres 4854   -->wf 5596   ` cfv 5600    <_ cle 9682   2c2 10665   #chash 12520   UMGrph cumg 25035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4545  ax-nul 4554  ax-pr 4659
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-br 4423  df-opab 4482  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-fun 5602  df-fn 5603  df-f 5604  df-umgra 25036
This theorem is referenced by:  eupares  25699  eupath2lem3  25703
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