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Theorem usgra0v 23443
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 23423 . . 3  |-  ( (/) USGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isusgra 23425 . . . 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 4129 . . . . . . . . . . . 12  |-  ~P (/)  =  { (/)
}
65difeq1i 3579 . . . . . . . . . . 11  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
7 difid 3856 . . . . . . . . . . 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 3073 . . . . . . . 8  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  (/)  |  (
# `  x )  =  2 } )
12 rab0 3767 . . . . . . . 8  |-  { x  e.  (/)  |  ( # `  x )  =  2 }  =  (/)
1311, 12syl6eq 2511 . . . . . . 7  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  (/) )
143, 4, 13f1eq123d 5745 . . . . . 6  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
E : dom  E -1-1-> (/) ) )
15 f1f 5715 . . . . . . 7  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
16 f00 5702 . . . . . . . 8  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
1716simplbi 460 . . . . . . 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 466 . . . 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 4531 . . . 4  |-  (/)  e.  _V
24 usgra0 23442 . . . 4  |-  ( (/)  e.  _V  ->  (/) USGrph  (/) )
2523, 24ax-mp 5 . . 3  |-  (/) USGrph  (/)
26 breq2 4405 . . 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 369    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    \ cdif 3434   (/)c0 3746   ~Pcpw 3969   {csn 3986   class class class wbr 4401   dom cdm 4949   -->wf 5523   -1-1->wf1 5524   ` cfv 5527   2c2 10483   #chash 12221   USGrph cusg 23417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-usgra 23419
This theorem is referenced by:  usgra1v  23461  usgrafisindb0  23474  frgra0v  30734
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