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Theorem usgrarnedg 23302
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 23276 . . . 4  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5446 . . . . 5  |-  ( Fun 
E  <->  E  Fn  dom  E )
32biimpi 194 . . . 4  |-  ( Fun 
E  ->  E  Fn  dom  E )
4 fvelrnb 5738 . . . 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 23275 . . . . 5  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
7 f1f 5605 . . . . 5  |-  ( E : dom  E -1-1-> {
y  e.  ~P V  |  ( # `  y
)  =  2 }  ->  E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
8 ffvelrn 5840 . . . . . . 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 2502 . . . . . . . 8  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  <->  Y  e.  { y  e.  ~P V  |  ( # `  y
)  =  2 } ) )
10 fveq2 5690 . . . . . . . . . . 11  |-  ( y  =  Y  ->  ( # `
 y )  =  ( # `  Y
) )
1110eqeq1d 2450 . . . . . . . . . 10  |-  ( y  =  Y  ->  (
( # `  y )  =  2  <->  ( # `  Y
)  =  2 ) )
1211elrab 3116 . . . . . . . . 9  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  <->  ( Y  e.  ~P V  /\  ( # `
 Y )  =  2 ) )
13 hash2prde 12178 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a E. b
( a  =/=  b  /\  Y  =  {
a ,  b } ) )
14 eleq1 2502 . . . . . . . . . . . . . . . . 17  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  <->  { a ,  b }  e.  ~P V ) )
15 prex 4533 . . . . . . . . . . . . . . . . . . 19  |-  { a ,  b }  e.  _V
1615elpw 3865 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b }  e.  ~P V  <->  { a ,  b }  C_  V )
17 vex 2974 . . . . . . . . . . . . . . . . . . . 20  |-  a  e. 
_V
18 vex 2974 . . . . . . . . . . . . . . . . . . . 20  |-  b  e. 
_V
1917, 18prss 4026 . . . . . . . . . . . . . . . . . . 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 1678 . . . . . . . . . . . 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 2752 . . . . . . . . . . . 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 2839 . . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2605   E.wrex 2715   {crab 2718    C_ wss 3327   ~Pcpw 3859   {cpr 3878   class class class wbr 4291   dom cdm 4839   ran crn 4840   Fun wfun 5411    Fn wfn 5412   -->wf 5413   -1-1->wf1 5414   ` cfv 5417   2c2 10370   #chash 12102   USGrph cusg 23263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-hash 12103  df-usgra 23265
This theorem is referenced by:  usgraedg3  23303  usgrarnedg1  23306  usgrasscusgra  23390  sizeusglecusglem1  23391
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