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Theorem usgrares 23111
Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 23081. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
usgrares  |-  ( V USGrph  E  ->  V USGrph  ( E  |`  A ) )

Proof of Theorem usgrares
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgraf 23097 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 resss 5122 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5026 . . . . 5  |-  ( ( E  |`  A )  C_  E  ->  dom  ( E  |`  A )  C_  dom  E )
42, 3mp1i 12 . . . 4  |-  ( V USGrph  E  ->  dom  ( E  |`  A )  C_  dom  E )
5 f1ssres 5601 . . . 4  |-  ( ( E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  /\  dom  ( E  |`  A )  C_  dom  E )  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
61, 4, 5syl2anc 654 . . 3  |-  ( V USGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } )
7 resdmres 5317 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
8 f1eq1 5589 . . . 4  |-  ( ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )  ->  ( ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( E  |`  A ) : dom  ( E  |`  A ) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
97, 8ax-mp 5 . . 3  |-  ( ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
( E  |`  A ) : dom  ( E  |`  A ) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
106, 9sylib 196 . 2  |-  ( V USGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A )
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
11 usgrav 23093 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
12 resexg 5137 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1312anim2i 564 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  ( E  |`  A )  e.  _V ) )
14 isusgra 23095 . . 3  |-  ( ( V  e.  _V  /\  ( E  |`  A )  e.  _V )  -> 
( V USGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A )
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
1511, 13, 143syl 20 . 2  |-  ( V USGrph  E  ->  ( V USGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A )
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
1610, 15mpbird 232 1  |-  ( V USGrph  E  ->  V USGrph  ( E  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   {crab 2709   _Vcvv 2962    \ cdif 3313    C_ wss 3316   (/)c0 3625   ~Pcpw 3848   {csn 3865   class class class wbr 4280   dom cdm 4827    |` cres 4829   -1-1->wf1 5403   ` cfv 5406   2c2 10359   #chash 12087   USGrph cusg 23087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-usgra 23089
This theorem is referenced by: (None)
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