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Theorem usgra0v 24573
Description: The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
Assertion
Ref Expression
usgra0v  |-  ( (/) USGrph  E  <-> 
E  =  (/) )

Proof of Theorem usgra0v
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgrav 24540 . . 3  |-  ( (/) USGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isusgra 24546 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) USGrph  E 
<->  E : dom  E -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
3 eqidd 2455 . . . . . . 7  |-  ( E  e.  _V  ->  E  =  E )
4 eqidd 2455 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  =  dom  E )
5 pw0 4163 . . . . . . . . . . . 12  |-  ~P (/)  =  { (/)
}
65difeq1i 3604 . . . . . . . . . . 11  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
7 difid 3884 . . . . . . . . . . 11  |-  ( {
(/) }  \  { (/) } )  =  (/)
86, 7eqtri 2483 . . . . . . . . . 10  |-  ( ~P (/)  \  { (/) } )  =  (/)
98a1i 11 . . . . . . . . 9  |-  ( E  e.  _V  ->  ( ~P (/)  \  { (/) } )  =  (/) )
10 biidd 237 . . . . . . . . 9  |-  ( E  e.  _V  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
119, 10rabeqbidv 3101 . . . . . . . 8  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  (/)  |  (
# `  x )  =  2 } )
12 rab0 3805 . . . . . . . 8  |-  { x  e.  (/)  |  ( # `  x )  =  2 }  =  (/)
1311, 12syl6eq 2511 . . . . . . 7  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  (/) )
143, 4, 13f1eq123d 5793 . . . . . 6  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
E : dom  E -1-1-> (/) ) )
15 f1f 5763 . . . . . . 7  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
16 f00 5749 . . . . . . . 8  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
1716simplbi 458 . . . . . . 7  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
1815, 17syl 16 . . . . . 6  |-  ( E : dom  E -1-1-> (/)  ->  E  =  (/) )
1914, 18syl6bi 228 . . . . 5  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
2019adantl 464 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
212, 20sylbid 215 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) USGrph  E  ->  E  =  (/) ) )
221, 21mpcom 36 . 2  |-  ( (/) USGrph  E  ->  E  =  (/) )
23 0ex 4569 . . . 4  |-  (/)  e.  _V
24 usgra0 24572 . . . 4  |-  ( (/)  e.  _V  ->  (/) USGrph  (/) )
2523, 24ax-mp 5 . . 3  |-  (/) USGrph  (/)
26 breq2 4443 . . 3  |-  ( E  =  (/)  ->  ( (/) USGrph  E  <->  (/) USGrph  (/) ) )
2725, 26mpbiri 233 . 2  |-  ( E  =  (/)  ->  (/) USGrph  E )
2822, 27impbii 188 1  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
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   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   dom cdm 4988   -->wf 5566   -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-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-usgra 24535
This theorem is referenced by:  usgra1v  24592  usgrafisindb0  24610  frgra0v  25195  usgo0s0  32807  usgo0s0ALT  32808
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