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Theorem usgrarnedg 23125
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 23099 . . . 4  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5435 . . . . 5  |-  ( Fun 
E  <->  E  Fn  dom  E )
32biimpi 194 . . . 4  |-  ( Fun 
E  ->  E  Fn  dom  E )
4 fvelrnb 5727 . . . 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 23098 . . . . 5  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
7 f1f 5594 . . . . 5  |-  ( E : dom  E -1-1-> {
y  e.  ~P V  |  ( # `  y
)  =  2 }  ->  E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
8 ffvelrn 5829 . . . . . . 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 2493 . . . . . . . 8  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  <->  Y  e.  { y  e.  ~P V  |  ( # `  y
)  =  2 } ) )
10 fveq2 5679 . . . . . . . . . . 11  |-  ( y  =  Y  ->  ( # `
 y )  =  ( # `  Y
) )
1110eqeq1d 2441 . . . . . . . . . 10  |-  ( y  =  Y  ->  (
( # `  y )  =  2  <->  ( # `  Y
)  =  2 ) )
1211elrab 3106 . . . . . . . . 9  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  <->  ( Y  e.  ~P V  /\  ( # `
 Y )  =  2 ) )
13 hash2prde 12162 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a E. b
( a  =/=  b  /\  Y  =  {
a ,  b } ) )
14 eleq1 2493 . . . . . . . . . . . . . . . . 17  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  <->  { a ,  b }  e.  ~P V ) )
15 prex 4522 . . . . . . . . . . . . . . . . . . 19  |-  { a ,  b }  e.  _V
1615elpw 3854 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b }  e.  ~P V  <->  { a ,  b }  C_  V )
17 vex 2965 . . . . . . . . . . . . . . . . . . . 20  |-  a  e. 
_V
18 vex 2965 . . . . . . . . . . . . . . . . . . . 20  |-  b  e. 
_V
1917, 18prss 4015 . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  ( Y  e. 
~P V  ->  (
a  e.  V  /\  b  e.  V )
) )
2423imdistanri 684 . . . . . . . . . . . . . 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 1677 . . . . . . . . . . . 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 2743 . . . . . . . . . . . 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 462 . . . . . . . . . 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 2830 . . 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 1362   E.wex 1589    e. wcel 1755    =/= wne 2596   E.wrex 2706   {crab 2709    C_ wss 3316   ~Pcpw 3848   {cpr 3867   class class class wbr 4280   dom cdm 4827   ran crn 4828   Fun wfun 5400    Fn wfn 5401   -->wf 5402   -1-1->wf1 5403   ` cfv 5406   2c2 10358   #chash 12086   USGrph cusg 23086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-2 10367  df-n0 10567  df-z 10634  df-uz 10849  df-fz 11424  df-hash 12087  df-usgra 23088
This theorem is referenced by:  usgraedg3  23126  usgrarnedg1  23129  usgrasscusgra  23213  sizeusglecusglem1  23214
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