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Theorem frgrancvvdeq 30661
Description: In a finite 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
frgrancvvdeq  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  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 frgrancvvdeq
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeqlem9 30660 . . 3  |-  ( 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 ) ) )
21adantr 465 . 2  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  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 ) ) )
3 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 )  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
43imp 429 . . . . . . 7  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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
) )  ->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) )
5 frisusgra 30610 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
65adantr 465 . . . . . . . . . . . . . 14  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  V USGrph  E )
76adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  V USGrph  E )
8 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  x  e.  V )
9 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  E  e.  Fin )
109adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  E  e.  Fin )
117, 8, 103jca 1168 . . . . . . . . . . . 12  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  ( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin ) )
1211adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin )
)
13 nbusgrafi 23379 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  x  e.  V  /\  E  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  x
)  e.  Fin )
1412, 13syl 16 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( <. V ,  E >. Neighbors  x )  e.  Fin )
157adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  V USGrph  E )
16 eldifi 3499 . . . . . . . . . . . . 13  |-  ( y  e.  ( V  \  { x } )  ->  y  e.  V
)
1716adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
y  e.  V )
1810adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  E  e.  Fin )
1915, 17, 183jca 1168 . . . . . . . . . . 11  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( V USGrph  E  /\  y  e.  V  /\  E  e.  Fin )
)
20 nbusgrafi 23379 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  y  e.  V  /\  E  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  y
)  e.  Fin )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( <. V ,  E >. Neighbors 
y )  e.  Fin )
22 hasheqf1o 12141 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. Neighbors  x )  e.  Fin  /\  ( <. V ,  E >. Neighbors 
y )  e.  Fin )  ->  ( ( # `  ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  y
) )  <->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
2314, 21, 22syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( ( # `  ( <. V ,  E >. Neighbors  x
) )  =  (
# `  ( <. V ,  E >. Neighbors  y ) )  <->  E. f  f : ( <. V ,  E >. Neighbors  x ) -1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
2423adantr 465 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 ) ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  y
) )  <->  E. f 
f : ( <. V ,  E >. Neighbors  x
)
-1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
2524adantr 465 . . . . . . 7  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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 ) )  <->  E. f  f : ( <. V ,  E >. Neighbors  x ) -1-1-onto-> ( <. V ,  E >. Neighbors 
y ) ) )
264, 25mpbird 232 . . . . . 6  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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
) ) )
278adantr 465 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  x  e.  V )
2815, 27, 183jca 1168 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin )
)
2928adantr 465 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 ) ) )  ->  ( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin ) )
3029adantr 465 . . . . . . 7  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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  /\  E  e. 
Fin ) )
31 hashnbgravd 23602 . . . . . . 7  |-  ( ( V USGrph  E  /\  x  e.  V  /\  E  e. 
Fin )  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( ( V VDeg  E
) `  x )
)
3230, 31syl 16 . . . . . 6  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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 )
)
3319adantr 465 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 ) ) )  ->  ( V USGrph  E  /\  y  e.  V  /\  E  e.  Fin ) )
3433adantr 465 . . . . . . 7  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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  /\  E  e. 
Fin ) )
35 hashnbgravd 23602 . . . . . . 7  |-  ( ( V USGrph  E  /\  y  e.  V  /\  E  e. 
Fin )  ->  ( # `
 ( <. V ,  E >. Neighbors  y ) )  =  ( ( V VDeg  E
) `  y )
)
3634, 35syl 16 . . . . . 6  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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 )
)
3726, 32, 363eqtr3d 2483 . . . . 5  |-  ( ( ( ( ( ( V FriendGrph  E  /\  E  e. 
Fin )  /\  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 ) )
3837exp31 604 . . . 4  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 ) ) ) )
3938ralimdva 2815 . . 3  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  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 ) ) ) )
4039ralimdva 2815 . 2  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  ( 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 ) ) ) )
412, 40mpd 15 1  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  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    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    e/ wnel 2621   A.wral 2736    \ cdif 3346   {csn 3898   <.cop 3904   class class class wbr 4313   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   Fincfn 7331   #chash 12124   USGrph cusg 23286   Neighbors cnbgra 23351   VDeg cvdg 23585   FriendGrph cfrgra 30606
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-xadd 11111  df-fz 11459  df-hash 12125  df-usgra 23288  df-nbgra 23354  df-vdgr 23586  df-frgra 30607
This theorem is referenced by: (None)
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