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Theorem frgrawopreglem4 25787
Description: Lemma 4 for frgrawopreg 25789. 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 25786 . . 3  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )
)
4 frgrancvvdgeq 25783 . . . 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 3195 . . . . . . . . 9  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
65, 1eleq2s 2549 . . . . . . . 8  |-  ( a  e.  A  ->  a  e.  V )
7 sneq 3980 . . . . . . . . . . 11  |-  ( x  =  a  ->  { x }  =  { a } )
87difeq2d 3553 . . . . . . . . . 10  |-  ( x  =  a  ->  ( V  \  { x }
)  =  ( V 
\  { a } ) )
9 oveq2 6303 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
) )
10 neleq2 2732 . . . . . . . . . . . 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 5870 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 a ) )
1312eqeq1d 2455 . . . . . . . . . . 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 3003 . . . . . . . . 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 3148 . . . . . . . 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 2523 . . . . . . . . . 10  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
20 eldif 3416 . . . . . . . . . 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 761 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  V )
23 eleq1a 2526 . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . 12  |-  ( b  e.  { a }  <-> 
b  =  a )
2826, 27sylnibr 307 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  e.  { a } )
2922, 28eldifd 3417 . . . . . . . . . 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 2731 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  e/  ( <. V ,  E >. Neighbors  a )  <-> 
b  e/  ( <. V ,  E >. Neighbors  a ) ) )
34 fveq2 5870 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( V VDeg  E ) `  y )  =  ( ( V VDeg  E ) `
 b ) )
3534eqeq2d 2463 . . . . . . . . 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 3148 . . . . . . 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 2735 . . . . . . . . 9  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  <->  b  e.  (
<. V ,  E >. Neighbors  a
) )
40 frisusgra 25732 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
41 nbgraeledg 25170 . . . . . . . . . . . . . . 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 4053 . . . . . . . . . . . . . . 15  |-  { b ,  a }  =  { a ,  b }
4443eleq1i 2522 . . . . . . . . . . . . . 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 26 . . . . . . . . . . 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 26 . . . . . . . . 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 2636 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
5251a1d 26 . . . . . . . . 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 26 . . . . . . . 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 32 . . . . . 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 57 . . . . 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 84 . . . 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 37 . . 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 43 . 2  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  { a ,  b }  e.  ran  E ) )
6059ralrimivv 2810 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 1446    e. wcel 1889    =/= wne 2624    e/ wnel 2625   A.wral 2739   {crab 2743    \ cdif 3403   {csn 3970   {cpr 3972   <.cop 3976   class class class wbr 4405   ran crn 4838   ` cfv 5585  (class class class)co 6295   USGrph cusg 25069   Neighbors cnbgra 25157   VDeg cvdg 25633   FriendGrph cfrgra 25728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-xadd 11417  df-fz 11792  df-hash 12523  df-usgra 25072  df-nbgra 25160  df-vdgr 25634  df-frgra 25729
This theorem is referenced by:  frgrawopreglem5  25788  frgrawopreg1  25790  frgrawopreg2  25791
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