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Theorem usgrarnedg 24362
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 24327 . . . 4  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5607 . . . . 5  |-  ( Fun 
E  <->  E  Fn  dom  E )
32biimpi 194 . . . 4  |-  ( Fun 
E  ->  E  Fn  dom  E )
4 fvelrnb 5905 . . . 4  |-  ( E  Fn  dom  E  -> 
( Y  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
51, 3, 43syl 20 . . 3  |-  ( V USGrph  E  ->  ( Y  e. 
ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
6 usgraf0 24326 . . . . 5  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
7 f1f 5771 . . . . 5  |-  ( E : dom  E -1-1-> {
y  e.  ~P V  |  ( # `  y
)  =  2 }  ->  E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
8 ffvelrn 6014 . . . . . . 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 2515 . . . . . . . 8  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  <->  Y  e.  { y  e.  ~P V  |  ( # `  y
)  =  2 } ) )
10 fveq2 5856 . . . . . . . . . . 11  |-  ( y  =  Y  ->  ( # `
 y )  =  ( # `  Y
) )
1110eqeq1d 2445 . . . . . . . . . 10  |-  ( y  =  Y  ->  (
( # `  y )  =  2  <->  ( # `  Y
)  =  2 ) )
1211elrab 3243 . . . . . . . . 9  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  <->  ( Y  e.  ~P V  /\  ( # `
 Y )  =  2 ) )
13 hash2prde 12498 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a E. b
( a  =/=  b  /\  Y  =  {
a ,  b } ) )
14 eleq1 2515 . . . . . . . . . . . . . . . . 17  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  <->  { a ,  b }  e.  ~P V ) )
15 prex 4679 . . . . . . . . . . . . . . . . . . 19  |-  { a ,  b }  e.  _V
1615elpw 4003 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b }  e.  ~P V  <->  { a ,  b }  C_  V )
17 vex 3098 . . . . . . . . . . . . . . . . . . . 20  |-  a  e. 
_V
18 vex 3098 . . . . . . . . . . . . . . . . . . . 20  |-  b  e. 
_V
1917, 18prss 4169 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 15  |-  ( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  ( Y  e. 
~P V  ->  (
a  e.  V  /\  b  e.  V )
) )
2423imdistanri 691 . . . . . . . . . . . . . 14  |-  ( ( Y  e.  ~P V  /\  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2524ex 434 . . . . . . . . . . . . 13  |-  ( Y  e.  ~P V  -> 
( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
26252eximdv 1699 . . . . . . . . . . . 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 2966 . . . . . . . . . . . 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 465 . . . . . . . . . 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 30 . . . . . 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 434 . . . . 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 20 . . . 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 2933 . . 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 429 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 369    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   E.wrex 2794   {crab 2797    C_ wss 3461   ~Pcpw 3997   {cpr 4016   class class class wbr 4437   dom cdm 4989   ran crn 4990   Fun wfun 5572    Fn wfn 5573   -->wf 5574   -1-1->wf1 5575   ` cfv 5578   2c2 10592   #chash 12387   USGrph cusg 24308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311
This theorem is referenced by:  edgprvtx  24363  usgraedg3  24364  usgrarnedg1  24367  usgrasscusgra  24461  sizeusglecusglem1  24462
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