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Theorem frgrancvvdeq 25163
Description: In a finite friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 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 25162 . . 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 463 . 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 459 . . . . . . . 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 427 . . . . . . 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 25113 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
65adantr 463 . . . . . . . . . . . . . 14  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  V USGrph  E )
76adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  V USGrph  E )
8 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  x  e.  V )
9 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( V FriendGrph  E  /\  E  e. 
Fin )  ->  E  e.  Fin )
109adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  E  e.  Fin )
117, 8, 103jca 1174 . . . . . . . . . . . 12  |-  ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  ->  ( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin ) )
1211adantr 463 . . . . . . . . . . 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 24569 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  V USGrph  E )
16 eldifi 3540 . . . . . . . . . . . . 13  |-  ( y  e.  ( V  \  { x } )  ->  y  e.  V
)
1716adantl 464 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
y  e.  V )
1810adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  E  e.  Fin )
1915, 17, 183jca 1174 . . . . . . . . . . 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 24569 . . . . . . . . . . 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 12324 . . . . . . . . . 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 659 . . . . . . . . 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 463 . . . . . . . 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 463 . . . . . . 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 463 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  ->  x  e.  V )
2815, 27, 183jca 1174 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  E  e.  Fin )  /\  x  e.  V
)  /\  y  e.  ( V  \  { x } ) )  -> 
( V USGrph  E  /\  x  e.  V  /\  E  e.  Fin )
)
2928adantr 463 . . . . . . . 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 463 . . . . . . 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 25033 . . . . . . 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 463 . . . . . . . 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 463 . . . . . . 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 25033 . . . . . . 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 2431 . . . . 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 602 . . . 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 2790 . . 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 2790 . 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826    e/ wnel 2578   A.wral 2732    \ cdif 3386   {csn 3944   <.cop 3950   class class class wbr 4367   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   Fincfn 7435   #chash 12307   USGrph cusg 24451   Neighbors cnbgra 24538   VDeg cvdg 25014   FriendGrph cfrgra 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-xadd 11240  df-fz 11594  df-hash 12308  df-usgra 24454  df-nbgra 24541  df-vdgr 25015  df-frgra 25110
This theorem is referenced by: (None)
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