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Theorem usgrares 21342
Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 21312. (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 21328 . . . 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 5028 . . . . 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 5605 . . . 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 643 . . 3  |-  ( V USGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } )
7 resdmres 5320 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
8 f1eq1 5593 . . . 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 8 . . 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 189 . 2  |-  ( V USGrph  E  ->  ( E  |`  A ) : dom  ( E  |`  A )
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
11 usgrav 21324 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
12 resexg 5144 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1312anim2i 553 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  ( E  |`  A )  e.  _V ) )
14 isusgra 21326 . . 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 19 . 2  |-  ( V USGrph  E  ->  ( V USGrph  ( E  |`  A )  <->  ( E  |`  A ) : dom  ( E  |`  A )
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
1610, 15mpbird 224 1  |-  ( V USGrph  E  ->  V USGrph  ( E  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172   dom cdm 4837    |` cres 4839   -1-1->wf1 5410   ` cfv 5413   2c2 10005   #chash 11573   USGrph cusg 21318
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-usgra 21320
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