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Theorem ausisusgra 23430
Description: The equivalence of the definitions of an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 28-Aug-2017.)
Hypothesis
Ref Expression
ausgra.1  |-  G  =  { <. v ,  e
>.  |  e  C_  { x  e.  ~P v  |  ( # `  x
)  =  2 } }
Assertion
Ref Expression
ausisusgra  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E  <-> 
V USGrph  (  _I  |`  E ) ) )
Distinct variable groups:    v, e, x, E    e, V, v, x    x, X    x, Y
Allowed substitution hints:    G( x, v, e)    X( v, e)    Y( v, e)

Proof of Theorem ausisusgra
StepHypRef Expression
1 ausgra.1 . . 3  |-  G  =  { <. v ,  e
>.  |  e  C_  { x  e.  ~P v  |  ( # `  x
)  =  2 } }
21isausgra 23429 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E  <-> 
E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
3 f1oi 5783 . . . 4  |-  (  _I  |`  E ) : E -1-1-onto-> E
4 dff1o5 5757 . . . . 5  |-  ( (  _I  |`  E ) : E -1-1-onto-> E  <->  ( (  _I  |`  E ) : E -1-1-> E  /\  ran  (  _I  |`  E )  =  E ) )
5 f1ss 5718 . . . . . . . . 9  |-  ( ( (  _I  |`  E ) : E -1-1-> E  /\  E  C_  { x  e. 
~P V  |  (
# `  x )  =  2 } )  ->  (  _I  |`  E ) : E -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 } )
6 dmresi 5268 . . . . . . . . . . 11  |-  dom  (  _I  |`  E )  =  E
76eqcomi 2467 . . . . . . . . . 10  |-  E  =  dom  (  _I  |`  E )
8 f1eq2 5709 . . . . . . . . . 10  |-  ( E  =  dom  (  _I  |`  E )  ->  (
(  _I  |`  E ) : E -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 }  <->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
97, 8ax-mp 5 . . . . . . . . 9  |-  ( (  _I  |`  E ) : E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 }  <-> 
(  _I  |`  E ) : dom  (  _I  |`  E ) -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 } )
105, 9sylib 196 . . . . . . . 8  |-  ( ( (  _I  |`  E ) : E -1-1-> E  /\  E  C_  { x  e. 
~P V  |  (
# `  x )  =  2 } )  ->  (  _I  |`  E ) : dom  (  _I  |`  E ) -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 } )
1110ex 434 . . . . . . 7  |-  ( (  _I  |`  E ) : E -1-1-> E  ->  ( E 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
1211a1d 25 . . . . . 6  |-  ( (  _I  |`  E ) : E -1-1-> E  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) ) )
1312adantr 465 . . . . 5  |-  ( ( (  _I  |`  E ) : E -1-1-> E  /\  ran  (  _I  |`  E )  =  E )  -> 
( ( V  e.  X  /\  E  e.  Y )  ->  ( E  C_  { x  e. 
~P V  |  (
# `  x )  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) ) )
144, 13sylbi 195 . . . 4  |-  ( (  _I  |`  E ) : E -1-1-onto-> E  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) ) )
153, 14ax-mp 5 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
16 f1f 5713 . . . . 5  |-  ( (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 }  ->  (  _I  |`  E ) : dom  (  _I  |`  E ) --> { x  e.  ~P V  |  (
# `  x )  =  2 } )
17 df-f 5529 . . . . . 6  |-  ( (  _I  |`  E ) : dom  (  _I  |`  E ) --> { x  e.  ~P V  |  ( # `  x
)  =  2 }  <-> 
( (  _I  |`  E )  Fn  dom  (  _I  |`  E )  /\  ran  (  _I  |`  E ) 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } ) )
18 rnresi 5289 . . . . . . . . . 10  |-  ran  (  _I  |`  E )  =  E
1918sseq1i 3487 . . . . . . . . 9  |-  ( ran  (  _I  |`  E ) 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 }  <->  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } )
2019biimpi 194 . . . . . . . 8  |-  ( ran  (  _I  |`  E ) 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 }  ->  E 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } )
2120a1d 25 . . . . . . 7  |-  ( ran  (  _I  |`  E ) 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 }  ->  ( ( V  e.  X  /\  E  e.  Y
)  ->  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
2221adantl 466 . . . . . 6  |-  ( ( (  _I  |`  E )  Fn  dom  (  _I  |`  E )  /\  ran  (  _I  |`  E ) 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } )  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  E  C_ 
{ x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2317, 22sylbi 195 . . . . 5  |-  ( (  _I  |`  E ) : dom  (  _I  |`  E ) --> { x  e.  ~P V  |  ( # `  x
)  =  2 }  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  E  C_ 
{ x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2416, 23syl 16 . . . 4  |-  ( (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 }  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  E  C_ 
{ x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2524com12 31 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( (  _I  |`  E ) : dom  (  _I  |`  E ) -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  E 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } ) )
2615, 25impbid 191 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  <->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
27 resiexg 6623 . . 3  |-  ( E  e.  Y  ->  (  _I  |`  E )  e. 
_V )
28 isusgra0 23426 . . . 4  |-  ( ( V  e.  X  /\  (  _I  |`  E )  e.  _V )  -> 
( V USGrph  (  _I  |`  E )  <->  (  _I  |`  E ) : dom  (  _I  |`  E )
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2928bicomd 201 . . 3  |-  ( ( V  e.  X  /\  (  _I  |`  E )  e.  _V )  -> 
( (  _I  |`  E ) : dom  (  _I  |`  E ) -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 }  <->  V USGrph  (  _I  |`  E ) ) )
3027, 29sylan2 474 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( (  _I  |`  E ) : dom  (  _I  |`  E ) -1-1-> { x  e.  ~P V  |  (
# `  x )  =  2 }  <->  V USGrph  (  _I  |`  E ) ) )
312, 26, 303bitrd 279 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E  <-> 
V USGrph  (  _I  |`  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2802   _Vcvv 3076    C_ wss 3435   ~Pcpw 3967   class class class wbr 4399   {copab 4456    _I cid 4738   dom cdm 4947   ran crn 4948    |` cres 4949    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525   2c2 10481   #chash 12219   USGrph cusg 23415
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-hash 12220  df-usgra 23417
This theorem is referenced by: (None)
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