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Theorem usgrares 23239
Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 23209. (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 23225 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 resss 5129 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5034 . . . . 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 5608 . . . 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 661 . . 3  |-  ( V USGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } )
7 resdmres 5324 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
8 f1eq1 5596 . . . 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 23221 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
12 resexg 5144 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1312anim2i 569 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  ( E  |`  A )  e.  _V ) )
14 isusgra 23223 . . 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 1369    e. wcel 1756   {crab 2714   _Vcvv 2967    \ cdif 3320    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   {csn 3872   class class class wbr 4287   dom cdm 4835    |` cres 4837   -1-1->wf1 5410   ` cfv 5413   2c2 10363   #chash 12095   USGrph cusg 23215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-usgra 23217
This theorem is referenced by: (None)
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