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Theorem usgra0v 21396
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 21376 . . 3  |-  ( (/) USGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isusgra 21378 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) USGrph  E 
<->  E : dom  E -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
3 eqidd 2439 . . . . . . 7  |-  ( E  e.  _V  ->  E  =  E )
4 eqidd 2439 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  =  dom  E )
5 pw0 3947 . . . . . . . . . . . 12  |-  ~P (/)  =  { (/)
}
65difeq1i 3463 . . . . . . . . . . 11  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
7 difid 3698 . . . . . . . . . . 11  |-  ( {
(/) }  \  { (/) } )  =  (/)
86, 7eqtri 2458 . . . . . . . . . 10  |-  ( ~P (/)  \  { (/) } )  =  (/)
98a1i 11 . . . . . . . . 9  |-  ( E  e.  _V  ->  ( ~P (/)  \  { (/) } )  =  (/) )
10 biidd 230 . . . . . . . . 9  |-  ( E  e.  _V  ->  (
( # `  x )  =  2  <->  ( # `  x
)  =  2 ) )
119, 10rabeqbidv 2953 . . . . . . . 8  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  (/)  |  (
# `  x )  =  2 } )
12 rab0 3650 . . . . . . . 8  |-  { x  e.  (/)  |  ( # `  x )  =  2 }  =  (/)
1311, 12syl6eq 2486 . . . . . . 7  |-  ( E  e.  _V  ->  { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 }  =  (/) )
143, 4, 13f1eq123d 5672 . . . . . 6  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  <-> 
E : dom  E -1-1-> (/) ) )
15 f1f 5642 . . . . . . 7  |-  ( E : dom  E -1-1-> (/)  ->  E : dom  E --> (/) )
16 f00 5631 . . . . . . . 8  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
1716simplbi 448 . . . . . . 7  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
1815, 17syl 16 . . . . . 6  |-  ( E : dom  E -1-1-> (/)  ->  E  =  (/) )
1914, 18syl6bi 221 . . . . 5  |-  ( E  e.  _V  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
2019adantl 454 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( E : dom  E -1-1-> {
x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  E  =  (/) ) )
212, 20sylbid 208 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) USGrph  E  ->  E  =  (/) ) )
221, 21mpcom 35 . 2  |-  ( (/) USGrph  E  ->  E  =  (/) )
23 0ex 4342 . . . 4  |-  (/)  e.  _V
24 usgra0 21395 . . . 4  |-  ( (/)  e.  _V  ->  (/) USGrph  (/) )
2523, 24ax-mp 5 . . 3  |-  (/) USGrph  (/)
26 breq2 4219 . . 3  |-  ( E  =  (/)  ->  ( (/) USGrph  E  <->  (/) USGrph  (/) ) )
2725, 26mpbiri 226 . 2  |-  ( E  =  (/)  ->  (/) USGrph  E )
2822, 27impbii 182 1  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    \ cdif 3319   (/)c0 3630   ~Pcpw 3801   {csn 3816   class class class wbr 4215   dom cdm 4881   -->wf 5453   -1-1->wf1 5454   ` cfv 5457   2c2 10054   #chash 11623   USGrph cusg 21370
This theorem is referenced by:  usgra1v  21414  usgrafisindb0  21427  frgra0v  28457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-usgra 21372
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