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Theorem frgrancvvdgeq 30634
Description: In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to the first Lemma ("claim") of the proof of the (friendship) theorem in [Huneke] p. 1: "If x,y, are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Assertion
Ref Expression
frgrancvvdgeq  |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) ) )
Distinct variable groups:    x, E, y    x, V, y

Proof of Theorem frgrancvvdgeq
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeqlem9 30632 . 2  |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
2 ovex 6115 . . . . . . . . . 10  |-  ( <. V ,  E >. Neighbors  x
)  e.  _V
3 ovex 6115 . . . . . . . . . 10  |-  ( <. V ,  E >. Neighbors  y
)  e.  _V
42, 3pm3.2i 455 . . . . . . . . 9  |-  ( (
<. V ,  E >. Neighbors  x
)  e.  _V  /\  ( <. V ,  E >. Neighbors 
y )  e.  _V )
5 hasheqf1oi 12121 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. Neighbors  x )  e.  _V  /\  ( <. V ,  E >. Neighbors 
y )  e.  _V )  ->  ( E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  y
) ) ) )
64, 5mp1i 12 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  ( E. f  f :
( <. V ,  E >. Neighbors  x ) -1-1-onto-> ( <. V ,  E >. Neighbors 
y )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  y
) ) ) )
76imim2d 52 . . . . . . 7  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  (
( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) )  -> 
( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( # `  ( <. V ,  E >. Neighbors  x
) )  =  (
# `  ( <. V ,  E >. Neighbors  y ) ) ) ) )
87imp31 432 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  y
) ) )
9 frisusgra 30582 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
109ad2antrr 725 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  V USGrph  E )
11 simplr 754 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  x  e.  V )
1210, 11jca 532 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  ( V USGrph  E  /\  x  e.  V ) )
1312ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  ( V USGrph  E  /\  x  e.  V ) )
14 hashnbgravdg 23580 . . . . . . 7  |-  ( ( V USGrph  E  /\  x  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( ( V VDeg  E
) `  x )
)
1513, 14syl 16 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( ( V VDeg  E
) `  x )
)
16 eldifi 3477 . . . . . . . . . 10  |-  ( y  e.  ( V  \  { x } )  ->  y  e.  V
)
1716adantl 466 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  y  e.  V )
1810, 17jca 532 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  ( V USGrph  E  /\  y  e.  V ) )
1918ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  ( V USGrph  E  /\  y  e.  V ) )
20 hashnbgravdg 23580 . . . . . . 7  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  y ) )  =  ( ( V VDeg  E
) `  y )
)
2119, 20syl 16 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  ( # `
 ( <. V ,  E >. Neighbors  y ) )  =  ( ( V VDeg  E
) `  y )
)
228, 15, 213eqtr3d 2482 . . . . 5  |-  ( ( ( ( ( V FriendGrph  E  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  /\  ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )  /\  y  e/  ( <. V ,  E >. Neighbors  x
) )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) )
2322exp31 604 . . . 4  |-  ( ( ( V FriendGrph  E  /\  x  e.  V )  /\  y  e.  ( V  \  { x }
) )  ->  (
( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) )  -> 
( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( ( V VDeg  E ) `  x
)  =  ( ( V VDeg  E ) `  y ) ) ) )
2423ralimdva 2793 . . 3  |-  ( ( V FriendGrph  E  /\  x  e.  V )  ->  ( A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) )  ->  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) ) ) )
2524ralimdva 2793 . 2  |-  ( V FriendGrph  E  ->  ( A. x  e.  V  A. y  e.  ( V  \  {
x } ) ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) )  ->  A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) ) ) )
261, 25mpd 15 1  |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    e/ wnel 2606   A.wral 2714   _Vcvv 2971    \ cdif 3324   {csn 3876   <.cop 3882   class class class wbr 4291   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6090   #chash 12102   USGrph cusg 23263   Neighbors cnbgra 23328   VDeg cvdg 23562   FriendGrph cfrgra 30578
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-xadd 11089  df-fz 11437  df-hash 12103  df-usgra 23265  df-nbgra 23331  df-vdgr 23563  df-frgra 30579
This theorem is referenced by:  frgrawopreglem4  30638
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