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Theorem frgrawopreglem4 25854
Description: Lemma 4 for frgrawopreg 25856. 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 25853 . . 3  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )
)
4 frgrancvvdgeq 25850 . . . 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 3181 . . . . . . . . 9  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
65, 1eleq2s 2567 . . . . . . . 8  |-  ( a  e.  A  ->  a  e.  V )
7 sneq 3969 . . . . . . . . . . 11  |-  ( x  =  a  ->  { x }  =  { a } )
87difeq2d 3540 . . . . . . . . . 10  |-  ( x  =  a  ->  ( V  \  { x }
)  =  ( V 
\  { a } ) )
9 oveq2 6316 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
) )
10 neleq2 2749 . . . . . . . . . . . 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 5879 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 a ) )
1312eqeq1d 2473 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
( ( V VDeg  E
) `  x )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  y ) ) )
1411, 13imbi12d 327 . . . . . . . . . 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 2987 . . . . . . . . 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 3132 . . . . . . . 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 472 . . . . . 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 2541 . . . . . . . . . 10  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
20 eldif 3400 . . . . . . . . . 10  |-  ( b  e.  ( V  \  A )  <->  ( b  e.  V  /\  -.  b  e.  A ) )
2119, 20bitri 257 . . . . . . . . 9  |-  ( b  e.  B  <->  ( b  e.  V  /\  -.  b  e.  A ) )
22 simpll 768 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  V )
23 eleq1a 2544 . . . . . . . . . . . . . . 15  |-  ( a  e.  A  ->  (
b  =  a  -> 
b  e.  A ) )
2423con3rr3 143 . . . . . . . . . . . . . 14  |-  ( -.  b  e.  A  -> 
( a  e.  A  ->  -.  b  =  a ) )
2524adantl 473 . . . . . . . . . . . . 13  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  -.  b  =  a ) )
2625imp 436 . . . . . . . . . . . 12  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  =  a )
27 elsn 3973 . . . . . . . . . . . 12  |-  ( b  e.  { a }  <-> 
b  =  a )
2826, 27sylnibr 312 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  e.  { a } )
2922, 28eldifd 3401 . . . . . . . . . 10  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  ( V  \  {
a } ) )
3029ex 441 . . . . . . . . 9  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  b  e.  ( V  \  {
a } ) ) )
3121, 30sylbi 200 . . . . . . . 8  |-  ( b  e.  B  ->  (
a  e.  A  -> 
b  e.  ( V 
\  { a } ) ) )
3231impcom 437 . . . . . . 7  |-  ( ( a  e.  A  /\  b  e.  B )  ->  b  e.  ( V 
\  { a } ) )
33 neleq1 2748 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  e/  ( <. V ,  E >. Neighbors  a )  <-> 
b  e/  ( <. V ,  E >. Neighbors  a ) ) )
34 fveq2 5879 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( V VDeg  E ) `  y )  =  ( ( V VDeg  E ) `
 b ) )
3534eqeq2d 2481 . . . . . . . . 9  |-  ( y  =  b  ->  (
( ( V VDeg  E
) `  a )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) )
3633, 35imbi12d 327 . . . . . . . 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 3132 . . . . . . 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 2752 . . . . . . . . 9  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  <->  b  e.  (
<. V ,  E >. Neighbors  a
) )
40 frisusgra 25799 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
41 nbgraeledg 25237 . . . . . . . . . . . . . . 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 4041 . . . . . . . . . . . . . . 15  |-  { b ,  a }  =  { a ,  b }
4443eleq1i 2540 . . . . . . . . . . . . . 14  |-  ( { b ,  a }  e.  ran  E  <->  { a ,  b }  e.  ran  E )
4542, 44syl6bb 269 . . . . . . . . . . . . 13  |-  ( V FriendGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors 
a )  <->  { a ,  b }  e.  ran  E ) )
4645biimpa 492 . . . . . . . . . . . 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 442 . . . . . . . . . 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 200 . . . . . . . 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 2654 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 a )  =  ( ( V VDeg  E
) `  b )  ->  ( ( ( V VDeg 
E ) `  a
)  =/=  ( ( V VDeg  E ) `  b )  ->  { a ,  b }  e.  ran  E ) )
5251a1d 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
) ) )
5352a1d 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 ) ) ) )
5450, 53ja 166 . . . . . . 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 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 ) ) ) )
5618, 38, 553syld 56 . . . . 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 83 . . . 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 36 . . 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 42 . 2  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  b  e.  B )  ->  { a ,  b }  e.  ran  E ) )
6059ralrimivv 2813 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    e/ wnel 2642   A.wral 2756   {crab 2760    \ cdif 3387   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224   VDeg cvdg 25700   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-xadd 11433  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-vdgr 25701  df-frgra 25796
This theorem is referenced by:  frgrawopreglem5  25855  frgrawopreg1  25857  frgrawopreg2  25858
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