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Theorem frgrawopreglem4 25771
Description: Lemma 4 for frgrawopreg 25773. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [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 25770 . . 3  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )
)
4 frgrancvvdgeq 25767 . . . 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 3227 . . . . . . . . 9  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
65, 1eleq2s 2531 . . . . . . . 8  |-  ( a  e.  A  ->  a  e.  V )
7 sneq 4008 . . . . . . . . . . 11  |-  ( x  =  a  ->  { x }  =  { a } )
87difeq2d 3585 . . . . . . . . . 10  |-  ( x  =  a  ->  ( V  \  { x }
)  =  ( V 
\  { a } ) )
9 oveq2 6312 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
) )
10 neleq2 2766 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
)  ->  ( y  e/  ( <. V ,  E >. Neighbors  x )  <->  y  e/  ( <. V ,  E >. Neighbors 
a ) ) )
119, 10syl 17 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
y  e/  ( <. V ,  E >. Neighbors  x )  <-> 
y  e/  ( <. V ,  E >. Neighbors  a ) ) )
12 fveq2 5880 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 a ) )
1312eqeq1d 2425 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
( ( V VDeg  E
) `  x )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  y ) ) )
1411, 13imbi12d 322 . . . . . . . . . 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 3040 . . . . . . . . 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 3179 . . . . . . . 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 17 . . . . . . 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 467 . . . . . 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 2501 . . . . . . . . . 10  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
20 eldif 3448 . . . . . . . . . 10  |-  ( b  e.  ( V  \  A )  <->  ( b  e.  V  /\  -.  b  e.  A ) )
2119, 20bitri 253 . . . . . . . . 9  |-  ( b  e.  B  <->  ( b  e.  V  /\  -.  b  e.  A ) )
22 simpll 759 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  V )
23 eleq1a 2506 . . . . . . . . . . . . . . 15  |-  ( a  e.  A  ->  (
b  =  a  -> 
b  e.  A ) )
2423con3rr3 142 . . . . . . . . . . . . . 14  |-  ( -.  b  e.  A  -> 
( a  e.  A  ->  -.  b  =  a ) )
2524adantl 468 . . . . . . . . . . . . 13  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  -.  b  =  a ) )
2625imp 431 . . . . . . . . . . . 12  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  =  a )
27 elsn 4012 . . . . . . . . . . . 12  |-  ( b  e.  { a }  <-> 
b  =  a )
2826, 27sylnibr 307 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  e.  { a } )
2922, 28eldifd 3449 . . . . . . . . . 10  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  ( V  \  {
a } ) )
3029ex 436 . . . . . . . . 9  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  b  e.  ( V  \  {
a } ) ) )
3121, 30sylbi 199 . . . . . . . 8  |-  ( b  e.  B  ->  (
a  e.  A  -> 
b  e.  ( V 
\  { a } ) ) )
3231impcom 432 . . . . . . 7  |-  ( ( a  e.  A  /\  b  e.  B )  ->  b  e.  ( V 
\  { a } ) )
33 neleq1 2764 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  e/  ( <. V ,  E >. Neighbors  a )  <-> 
b  e/  ( <. V ,  E >. Neighbors  a ) ) )
34 fveq2 5880 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( V VDeg  E ) `  y )  =  ( ( V VDeg  E ) `
 b ) )
3534eqeq2d 2437 . . . . . . . . 9  |-  ( y  =  b  ->  (
( ( V VDeg  E
) `  a )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) )
3633, 35imbi12d 322 . . . . . . . 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 3179 . . . . . . 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 17 . . . . . 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 2771 . . . . . . . . 9  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  <->  b  e.  (
<. V ,  E >. Neighbors  a
) )
40 frisusgra 25716 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
41 nbgraeledg 25154 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { b ,  a }  e.  ran  E ) )
4240, 41syl 17 . . . . . . . . . . . . . 14  |-  ( V FriendGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { b ,  a }  e.  ran  E ) )
43 prcom 4077 . . . . . . . . . . . . . . 15  |-  { b ,  a }  =  { a ,  b }
4443eleq1i 2500 . . . . . . . . . . . . . 14  |-  ( { b ,  a }  e.  ran  E  <->  { a ,  b }  e.  ran  E )
4542, 44syl6bb 265 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { a ,  b }  e.  ran  E ) )
4645biimpa 487 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  b  e.  ( <. V ,  E >. Neighbors 
a ) )  ->  { a ,  b }  e.  ran  E
)
4746a1d 27 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  b  e.  ( <. V ,  E >. Neighbors 
a ) )  -> 
( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
4847expcom 437 . . . . . . . . . 10  |-  ( b  e.  ( <. V ,  E >. Neighbors  a )  ->  ( V FriendGrph  E  ->  ( (
( V VDeg  E ) `  a )  =/=  (
( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) ) )
4948a1d 27 . . . . . . . . 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 199 . . . . . . . 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 eqneqall 2632 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
5251a1d 27 . . . . . . . . 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
) ) )
5352a1d 27 . . . . . . . 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 ) ) ) )
5450, 53ja 165 . . . . . . 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 ) ) ) )
5554com12 33 . . . . . 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 ) ) ) )
5618, 38, 553syld 58 . . . . 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
) ) ) )
5756com3l 85 . . . 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 ) ) ) )
584, 57mpcom 38 . . 3  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )  ->  { a ,  b }  e.  ran  E
) ) )
593, 58mpdi 44 . 2  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  { a ,  b }  e.  ran  E ) )
6059ralrimivv 2846 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 188    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619    e/ wnel 2620   A.wral 2776   {crab 2780    \ cdif 3435   {csn 3998   {cpr 4000   <.cop 4004   class class class wbr 4422   ran crn 4853   ` cfv 5600  (class class class)co 6304   USGrph cusg 25053   Neighbors cnbgra 25141   VDeg cvdg 25617   FriendGrph cfrgra 25712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596  ax-cnex 9601  ax-resscn 9602  ax-1cn 9603  ax-icn 9604  ax-addcl 9605  ax-addrcl 9606  ax-mulcl 9607  ax-mulrcl 9608  ax-mulcom 9609  ax-addass 9610  ax-mulass 9611  ax-distr 9612  ax-i2m1 9613  ax-1ne0 9614  ax-1rid 9615  ax-rnegex 9616  ax-rrecex 9617  ax-cnre 9618  ax-pre-lttri 9619  ax-pre-lttrn 9620  ax-pre-ltadd 9621  ax-pre-mulgt0 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-pss 3454  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4219  df-int 4255  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-tr 4518  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6266  df-ov 6307  df-oprab 6308  df-mpt2 6309  df-om 6706  df-1st 6806  df-2nd 6807  df-wrecs 7038  df-recs 7100  df-rdg 7138  df-1o 7192  df-2o 7193  df-oadd 7196  df-er 7373  df-en 7580  df-dom 7581  df-sdom 7582  df-fin 7583  df-card 8380  df-cda 8604  df-pnf 9683  df-mnf 9684  df-xr 9685  df-ltxr 9686  df-le 9687  df-sub 9868  df-neg 9869  df-nn 10616  df-2 10674  df-n0 10876  df-z 10944  df-uz 11166  df-xadd 11416  df-fz 11791  df-hash 12521  df-usgra 25056  df-nbgra 25144  df-vdgr 25618  df-frgra 25713
This theorem is referenced by:  frgrawopreglem5  25772  frgrawopreg1  25774  frgrawopreg2  25775
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