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Theorem ausisusgra 24482
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 24481 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E  <-> 
E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
3 f1oi 5857 . . . 4  |-  (  _I  |`  E ) : E -1-1-onto-> E
4 dff1o5 5831 . . . . 5  |-  ( (  _I  |`  E ) : E -1-1-onto-> E  <->  ( (  _I  |`  E ) : E -1-1-> E  /\  ran  (  _I  |`  E )  =  E ) )
5 f1ss 5792 . . . . . . . . 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 5339 . . . . . . . . . . 11  |-  dom  (  _I  |`  E )  =  E
76eqcomi 2470 . . . . . . . . . 10  |-  E  =  dom  (  _I  |`  E )
8 f1eq2 5783 . . . . . . . . . 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 5787 . . . . 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 5598 . . . . . 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 5360 . . . . . . . . . 10  |-  ran  (  _I  |`  E )  =  E
1918sseq1i 3523 . . . . . . . . 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 6735 . . 3  |-  ( E  e.  Y  ->  (  _I  |`  E )  e. 
_V )
28 isusgra0 24474 . . . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   class class class wbr 4456   {copab 4514    _I cid 4799   dom cdm 5008   ran crn 5009    |` cres 5010    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594   2c2 10606   #chash 12408   USGrph cusg 24457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460
This theorem is referenced by:  ausisusgraedg  24483
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