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Theorem usgra0v 24144
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 24111 . . 3  |-  ( (/) USGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isusgra 24117 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) USGrph  E 
<->  E : dom  E -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
3 eqidd 2468 . . . . . . 7  |-  ( E  e.  _V  ->  E  =  E )
4 eqidd 2468 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  =  dom  E )
5 pw0 4174 . . . . . . . . . . . 12  |-  ~P (/)  =  { (/)
}
65difeq1i 3618 . . . . . . . . . . 11  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
7 difid 3895 . . . . . . . . . . 11  |-  ( {
(/) }  \  { (/) } )  =  (/)
86, 7eqtri 2496 . . . . . . . . . 10  |-  ( ~P (/)  \  { (/) } )  =  (/)
98a1i 11 . . . . . . . . 9  |-  ( E  e.  _V  ->  ( ~P (/)  \  { (/) } )  =  (/) )
10 biidd 237 . . . . . . . . 9  |-  ( E  e.  _V  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
119, 10rabeqbidv 3108 . . . . . . . 8  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  (/)  |  (
# `  x )  =  2 } )
12 rab0 3806 . . . . . . . 8  |-  { x  e.  (/)  |  ( # `  x )  =  2 }  =  (/)
1311, 12syl6eq 2524 . . . . . . 7  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  (/) )
143, 4, 13f1eq123d 5811 . . . . . 6  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
E : dom  E -1-1-> (/) ) )
15 f1f 5781 . . . . . . 7  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
16 f00 5767 . . . . . . . 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 4577 . . . 4  |-  (/)  e.  _V
24 usgra0 24143 . . . 4  |-  ( (/)  e.  _V  ->  (/) USGrph  (/) )
2523, 24ax-mp 5 . . 3  |-  (/) USGrph  (/)
26 breq2 4451 . . 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 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   dom cdm 4999   -->wf 5584   -1-1->wf1 5585   ` cfv 5588   2c2 10586   #chash 12374   USGrph cusg 24103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-usgra 24106
This theorem is referenced by:  usgra1v  24163  usgrafisindb0  24181  frgra0v  24766  usgo0s0  32129  usgo0s0ALT  32130
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