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Theorem usgrarnedg 24801
Description: For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
Assertion
Ref Expression
usgrarnedg  |-  ( ( V USGrph  E  /\  Y  e. 
ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
Distinct variable groups:    E, a,
b    Y, a, b    V, a, b

Proof of Theorem usgrarnedg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrafun 24766 . . . 4  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5598 . . . . 5  |-  ( Fun 
E  <->  E  Fn  dom  E )
32biimpi 194 . . . 4  |-  ( Fun 
E  ->  E  Fn  dom  E )
4 fvelrnb 5896 . . . 4  |-  ( E  Fn  dom  E  -> 
( Y  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
51, 3, 43syl 18 . . 3  |-  ( V USGrph  E  ->  ( Y  e. 
ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
6 usgraf0 24765 . . . . 5  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
7 f1f 5764 . . . . 5  |-  ( E : dom  E -1-1-> {
y  e.  ~P V  |  ( # `  y
)  =  2 }  ->  E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
8 ffvelrn 6007 . . . . . . 7  |-  ( ( E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 }  /\  x  e.  dom  E )  ->  ( E `  x )  e.  {
y  e.  ~P V  |  ( # `  y
)  =  2 } )
9 eleq1 2474 . . . . . . . 8  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  <->  Y  e.  { y  e.  ~P V  |  ( # `  y
)  =  2 } ) )
10 fveq2 5849 . . . . . . . . . . 11  |-  ( y  =  Y  ->  ( # `
 y )  =  ( # `  Y
) )
1110eqeq1d 2404 . . . . . . . . . 10  |-  ( y  =  Y  ->  (
( # `  y )  =  2  <->  ( # `  Y
)  =  2 ) )
1211elrab 3207 . . . . . . . . 9  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  <->  ( Y  e.  ~P V  /\  ( # `
 Y )  =  2 ) )
13 hash2prde 12565 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a E. b
( a  =/=  b  /\  Y  =  {
a ,  b } ) )
14 eleq1 2474 . . . . . . . . . . . . . . . . 17  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  <->  { a ,  b }  e.  ~P V ) )
15 prex 4633 . . . . . . . . . . . . . . . . . . 19  |-  { a ,  b }  e.  _V
1615elpw 3961 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b }  e.  ~P V  <->  { a ,  b }  C_  V )
17 vex 3062 . . . . . . . . . . . . . . . . . . . 20  |-  a  e. 
_V
18 vex 3062 . . . . . . . . . . . . . . . . . . . 20  |-  b  e. 
_V
1917, 18prss 4126 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  V  /\  b  e.  V )  <->  { a ,  b } 
C_  V )
2019biimpri 206 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b } 
C_  V  ->  (
a  e.  V  /\  b  e.  V )
)
2116, 20sylbi 195 . . . . . . . . . . . . . . . . 17  |-  ( { a ,  b }  e.  ~P V  -> 
( a  e.  V  /\  b  e.  V
) )
2214, 21syl6bi 228 . . . . . . . . . . . . . . . 16  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  ->  ( a  e.  V  /\  b  e.  V
) ) )
2322adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  ( Y  e. 
~P V  ->  (
a  e.  V  /\  b  e.  V )
) )
2423imdistanri 689 . . . . . . . . . . . . . 14  |-  ( ( Y  e.  ~P V  /\  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2524ex 432 . . . . . . . . . . . . 13  |-  ( Y  e.  ~P V  -> 
( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
26252eximdv 1733 . . . . . . . . . . . 12  |-  ( Y  e.  ~P V  -> 
( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  (
a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
27 r2ex 2930 . . . . . . . . . . . 12  |-  ( E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } )  <->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2826, 27syl6ibr 227 . . . . . . . . . . 11  |-  ( Y  e.  ~P V  -> 
( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2928adantr 463 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  ( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3013, 29mpd 15 . . . . . . . . 9  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
3112, 30sylbi 195 . . . . . . . 8  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  ->  E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } ) )
329, 31syl6bi 228 . . . . . . 7  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  ->  E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } ) ) )
338, 32syl5com 28 . . . . . 6  |-  ( ( E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 }  /\  x  e.  dom  E )  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3433ex 432 . . . . 5  |-  ( E : dom  E --> { y  e.  ~P V  | 
( # `  y )  =  2 }  ->  ( x  e.  dom  E  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) ) ) )
356, 7, 343syl 18 . . . 4  |-  ( V USGrph  E  ->  ( x  e. 
dom  E  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
3635rexlimdv 2894 . . 3  |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `
 x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
375, 36sylbid 215 . 2  |-  ( V USGrph  E  ->  ( Y  e. 
ran  E  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3837imp 427 1  |-  ( ( V USGrph  E  /\  Y  e. 
ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   E.wrex 2755   {crab 2758    C_ wss 3414   ~Pcpw 3955   {cpr 3974   class class class wbr 4395   dom cdm 4823   ran crn 4824   Fun wfun 5563    Fn wfn 5564   -->wf 5565   -1-1->wf1 5566   ` cfv 5569   2c2 10626   #chash 12452   USGrph cusg 24747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-hash 12453  df-usgra 24750
This theorem is referenced by:  edgprvtx  24802  usgraedg3  24803  usgrarnedg1  24806  usgrasscusgra  24900  sizeusglecusglem1  24901
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