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Theorem frgrawopreglem4 30640
Description: Lemma 4 for frgrawopreg 30642. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to the observation in the proof of [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B." (Contributed by Alexander van der Vekens, 30-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
frgrawopreg.b  |-  B  =  ( V  \  A
)
Assertion
Ref Expression
frgrawopreglem4  |-  ( V FriendGrph  E  ->  A. a  e.  A  A. b  e.  B  { a ,  b }  e.  ran  E
)
Distinct variable groups:    x, A    x, E    x, K    x, V    A, b    x, a, b, E    V, a,
b
Allowed substitution hints:    A( a)    B( x, a, b)    K( a, b)

Proof of Theorem frgrawopreglem4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 frgrawopreg.a . . . 4  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
2 frgrawopreg.b . . . 4  |-  B  =  ( V  \  A
)
31, 2frgrawopreglem3 30639 . . 3  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )
)
4 frgrancvvdgeq 30636 . . . 4  |-  ( 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 ) ) )
5 elrabi 3114 . . . . . . . . 9  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
65, 1eleq2s 2535 . . . . . . . 8  |-  ( a  e.  A  ->  a  e.  V )
7 sneq 3887 . . . . . . . . . . 11  |-  ( x  =  a  ->  { x }  =  { a } )
87difeq2d 3474 . . . . . . . . . 10  |-  ( x  =  a  ->  ( V  \  { x }
)  =  ( V 
\  { a } ) )
9 oveq2 6099 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
) )
10 neleq2 2710 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
)  ->  ( y  e/  ( <. V ,  E >. Neighbors  x )  <->  y  e/  ( <. V ,  E >. Neighbors 
a ) ) )
119, 10syl 16 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
y  e/  ( <. V ,  E >. Neighbors  x )  <-> 
y  e/  ( <. V ,  E >. Neighbors  a ) ) )
12 fveq2 5691 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 a ) )
1312eqeq1d 2451 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
( ( V VDeg  E
) `  x )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  y ) ) )
1411, 13imbi12d 320 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( ( V VDeg  E ) `  x
)  =  ( ( V VDeg  E ) `  y ) )  <->  ( y  e/  ( <. V ,  E >. Neighbors 
a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 y ) ) ) )
158, 14raleqbidv 2931 . . . . . . . . 9  |-  ( x  =  a  ->  ( A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) )  <->  A. y  e.  ( V  \  { a } ) ( y  e/  ( <. V ,  E >. Neighbors 
a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 y ) ) ) )
1615rspcv 3069 . . . . . . . 8  |-  ( a  e.  V  ->  ( A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) )  ->  A. y  e.  ( V  \  { a } ) ( y  e/  ( <. V ,  E >. Neighbors  a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 y ) ) ) )
176, 16syl 16 . . . . . . 7  |-  ( a  e.  A  ->  ( A. x  e.  V  A. y  e.  ( V  \  { x }
) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 y ) )  ->  A. y  e.  ( V  \  { a } ) ( y  e/  ( <. V ,  E >. Neighbors  a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 y ) ) ) )
1817adantr 465 . . . . . 6  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( A. x  e.  V  A. y  e.  ( V  \  {
x } ) ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( ( V VDeg  E ) `  x
)  =  ( ( V VDeg  E ) `  y ) )  ->  A. y  e.  ( V  \  { a } ) ( y  e/  ( <. V ,  E >. Neighbors 
a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 y ) ) ) )
192eleq2i 2507 . . . . . . . . . 10  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
20 eldif 3338 . . . . . . . . . 10  |-  ( b  e.  ( V  \  A )  <->  ( b  e.  V  /\  -.  b  e.  A ) )
2119, 20bitri 249 . . . . . . . . 9  |-  ( b  e.  B  <->  ( b  e.  V  /\  -.  b  e.  A ) )
22 simpll 753 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  V )
23 eleq1a 2512 . . . . . . . . . . . . . . 15  |-  ( a  e.  A  ->  (
b  =  a  -> 
b  e.  A ) )
2423con3rr3 136 . . . . . . . . . . . . . 14  |-  ( -.  b  e.  A  -> 
( a  e.  A  ->  -.  b  =  a ) )
2524adantl 466 . . . . . . . . . . . . 13  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  -.  b  =  a ) )
2625imp 429 . . . . . . . . . . . 12  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  =  a )
27 elsn 3891 . . . . . . . . . . . 12  |-  ( b  e.  { a }  <-> 
b  =  a )
2826, 27sylnibr 305 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  e.  { a } )
2922, 28eldifd 3339 . . . . . . . . . 10  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  ( V  \  {
a } ) )
3029ex 434 . . . . . . . . 9  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  b  e.  ( V  \  {
a } ) ) )
3121, 30sylbi 195 . . . . . . . 8  |-  ( b  e.  B  ->  (
a  e.  A  -> 
b  e.  ( V 
\  { a } ) ) )
3231impcom 430 . . . . . . 7  |-  ( ( a  e.  A  /\  b  e.  B )  ->  b  e.  ( V 
\  { a } ) )
33 neleq1 2709 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  e/  ( <. V ,  E >. Neighbors  a )  <-> 
b  e/  ( <. V ,  E >. Neighbors  a ) ) )
34 fveq2 5691 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( V VDeg  E ) `  y )  =  ( ( V VDeg  E ) `
 b ) )
3534eqeq2d 2454 . . . . . . . . 9  |-  ( y  =  b  ->  (
( ( V VDeg  E
) `  a )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) )
3633, 35imbi12d 320 . . . . . . . 8  |-  ( y  =  b  ->  (
( y  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  y ) )  <->  ( b  e/  ( <. V ,  E >. Neighbors 
a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 b ) ) ) )
3736rspcv 3069 . . . . . . 7  |-  ( b  e.  ( V  \  { a } )  ->  ( A. y  e.  ( V  \  {
a } ) ( y  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  y ) )  -> 
( b  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) ) )
3832, 37syl 16 . . . . . 6  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( A. y  e.  ( V  \  {
a } ) ( y  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  y ) )  -> 
( b  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) ) )
39 nnel 2714 . . . . . . . . 9  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  <->  b  e.  (
<. V ,  E >. Neighbors  a
) )
40 frisusgra 30584 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
41 nbgraeledg 23341 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { b ,  a }  e.  ran  E ) )
4240, 41syl 16 . . . . . . . . . . . . . 14  |-  ( V FriendGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { b ,  a }  e.  ran  E ) )
43 prcom 3953 . . . . . . . . . . . . . . 15  |-  { b ,  a }  =  { a ,  b }
4443eleq1i 2506 . . . . . . . . . . . . . 14  |-  ( { b ,  a }  e.  ran  E  <->  { a ,  b }  e.  ran  E )
4542, 44syl6bb 261 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { a ,  b }  e.  ran  E ) )
4645biimpa 484 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  b  e.  ( <. V ,  E >. Neighbors 
a ) )  ->  { a ,  b }  e.  ran  E
)
4746a1d 25 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  b  e.  ( <. V ,  E >. Neighbors 
a ) )  -> 
( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
4847expcom 435 . . . . . . . . . 10  |-  ( b  e.  ( <. V ,  E >. Neighbors  a )  ->  ( V FriendGrph  E  ->  ( (
( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) )
4948a1d 25 . . . . . . . . 9  |-  ( b  e.  ( <. V ,  E >. Neighbors  a )  ->  (
( a  e.  A  /\  b  e.  B
)  ->  ( V FriendGrph  E  ->  ( ( ( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) ) )
5039, 49sylbi 195 . . . . . . . 8  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( (
a  e.  A  /\  b  e.  B )  ->  ( V FriendGrph  E  ->  ( ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )  ->  { a ,  b }  e.  ran  E
) ) ) )
51 df-ne 2608 . . . . . . . . . . 11  |-  ( ( ( V VDeg  E ) `
 a )  =/=  ( ( V VDeg  E
) `  b )  <->  -.  ( ( V VDeg  E
) `  a )  =  ( ( V VDeg 
E ) `  b
) )
52 pm2.24 109 . . . . . . . . . . 11  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( -.  ( ( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 b )  ->  { a ,  b }  e.  ran  E
) )
5351, 52syl5bi 217 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
5453a1d 25 . . . . . . . . 9  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( V FriendGrph  E  ->  ( ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )  ->  { a ,  b }  e.  ran  E
) ) )
5554a1d 25 . . . . . . . 8  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( ( a  e.  A  /\  b  e.  B )  ->  ( V FriendGrph  E  ->  ( (
( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) ) )
5650, 55ja 161 . . . . . . 7  |-  ( ( b  e/  ( <. V ,  E >. Neighbors  a
)  ->  ( ( V VDeg  E ) `  a
)  =  ( ( V VDeg  E ) `  b ) )  -> 
( ( a  e.  A  /\  b  e.  B )  ->  ( V FriendGrph  E  ->  ( (
( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) ) )
5756com12 31 . . . . . 6  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( b  e/  ( <. V ,  E >. Neighbors 
a )  ->  (
( V VDeg  E ) `  a )  =  ( ( V VDeg  E ) `
 b ) )  ->  ( V FriendGrph  E  -> 
( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) ) )
5818, 38, 573syld 55 . . . . 5  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( A. x  e.  V  A. y  e.  ( V  \  {
x } ) ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( ( V VDeg  E ) `  x
)  =  ( ( V VDeg  E ) `  y ) )  -> 
( V FriendGrph  E  ->  (
( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )  ->  { a ,  b }  e.  ran  E
) ) ) )
5958com3l 81 . . . 4  |-  ( A. x  e.  V  A. y  e.  ( V  \  { x } ) ( y  e/  ( <. V ,  E >. Neighbors  x
)  ->  ( ( V VDeg  E ) `  x
)  =  ( ( V VDeg  E ) `  y ) )  -> 
( V FriendGrph  E  ->  (
( a  e.  A  /\  b  e.  B
)  ->  ( (
( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) ) )
604, 59mpcom 36 . . 3  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )  ->  { a ,  b }  e.  ran  E
) ) )
613, 60mpdi 42 . 2  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  { a ,  b }  e.  ran  E ) )
6261ralrimivv 2807 1  |-  ( V FriendGrph  E  ->  A. a  e.  A  A. b  e.  B  { a ,  b }  e.  ran  E
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606    e/ wnel 2607   A.wral 2715   {crab 2719    \ cdif 3325   {csn 3877   {cpr 3879   <.cop 3883   class class class wbr 4292   ran crn 4841   ` cfv 5418  (class class class)co 6091   USGrph cusg 23264   Neighbors cnbgra 23329   VDeg cvdg 23563   FriendGrph cfrgra 30580
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-xadd 11090  df-fz 11438  df-hash 12104  df-usgra 23266  df-nbgra 23332  df-vdgr 23564  df-frgra 30581
This theorem is referenced by:  frgrawopreglem5  30641  frgrawopreg1  30643  frgrawopreg2  30644
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