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Theorem umgrares 24445
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 24438 . . . 4  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 resss 5209 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5115 . . . . 5  |-  ( ( E  |`  A )  C_  E  ->  dom  ( E  |`  A )  C_  dom  E )
42, 3mp1i 12 . . . 4  |-  ( V UMGrph  E  ->  dom  ( E  |`  A )  C_  dom  E )
51, 4fssresd 5660 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
6 resdmres 5406 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
76feq1i 5631 . . 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 196 . 2  |-  ( V UMGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
9 relumgra 24435 . . . 4  |-  Rel UMGrph
109brrelexi 4954 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
119brrelex2i 4955 . . . 4  |-  ( V UMGrph  E  ->  E  e.  _V )
12 resexg 5228 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1311, 12syl 16 . . 3  |-  ( V UMGrph  E  ->  ( E  |`  A )  e.  _V )
14 isumgra 24436 . . 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 659 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A ) --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } ) )
168, 15mpbird 232 1  |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1826   {crab 2736   _Vcvv 3034    \ cdif 3386    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   {csn 3944   class class class wbr 4367   dom cdm 4913    |` cres 4915   -->wf 5492   ` cfv 5496    <_ cle 9540   2c2 10502   #chash 12307   UMGrph cumg 24433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-fun 5498  df-fn 5499  df-f 5500  df-umgra 24434
This theorem is referenced by:  eupares  25096  eupath2lem3  25100
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