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Theorem usgrares 24571
Description: A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 24526. (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 24548 . . . 4  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
2 resss 5285 . . . . 5  |-  ( E  |`  A )  C_  E
3 dmss 5191 . . . . 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 5770 . . . 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 659 . . 3  |-  ( V USGrph  E  ->  ( E  |`  dom  ( E  |`  A ) ) : dom  ( E  |`  A ) -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } )
7 resdmres 5481 . . . 4  |-  ( E  |`  dom  ( E  |`  A ) )  =  ( E  |`  A )
8 f1eq1 5758 . . . 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 24540 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
12 resexg 5304 . . . 4  |-  ( E  e.  _V  ->  ( E  |`  A )  e. 
_V )
1312anim2i 567 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  ( E  |`  A )  e.  _V ) )
14 isusgra 24546 . . 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 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   dom cdm 4988    |` cres 4990   -1-1->wf1 5567   ` cfv 5570   2c2 10581   #chash 12387   USGrph cusg 24532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-usgra 24535
This theorem is referenced by: (None)
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