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Theorem frgrawopreglem4 25168
Description: Lemma 4 for frgrawopreg 25170. 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 25167 . . 3  |-  ( ( a  e.  A  /\  b  e.  B )  ->  ( ( V VDeg  E
) `  a )  =/=  ( ( V VDeg  E
) `  b )
)
4 frgrancvvdgeq 25164 . . . 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 3179 . . . . . . . . 9  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
65, 1eleq2s 2490 . . . . . . . 8  |-  ( a  e.  A  ->  a  e.  V )
7 sneq 3954 . . . . . . . . . . 11  |-  ( x  =  a  ->  { x }  =  { a } )
87difeq2d 3536 . . . . . . . . . 10  |-  ( x  =  a  ->  ( V  \  { x }
)  =  ( V 
\  { a } ) )
9 oveq2 6204 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  a
) )
10 neleq2 2722 . . . . . . . . . . . 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 5774 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
 a ) )
1312eqeq1d 2384 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
( ( V VDeg  E
) `  x )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  y ) ) )
1411, 13imbi12d 318 . . . . . . . . . 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 2993 . . . . . . . . 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 3131 . . . . . . . 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 463 . . . . . 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 2460 . . . . . . . . . 10  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
20 eldif 3399 . . . . . . . . . 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 751 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  V )
23 eleq1a 2465 . . . . . . . . . . . . . . 15  |-  ( a  e.  A  ->  (
b  =  a  -> 
b  e.  A ) )
2423con3rr3 136 . . . . . . . . . . . . . 14  |-  ( -.  b  e.  A  -> 
( a  e.  A  ->  -.  b  =  a ) )
2524adantl 464 . . . . . . . . . . . . 13  |-  ( ( b  e.  V  /\  -.  b  e.  A
)  ->  ( a  e.  A  ->  -.  b  =  a ) )
2625imp 427 . . . . . . . . . . . 12  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  =  a )
27 elsn 3958 . . . . . . . . . . . 12  |-  ( b  e.  { a }  <-> 
b  =  a )
2826, 27sylnibr 303 . . . . . . . . . . 11  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  -.  b  e.  { a } )
2922, 28eldifd 3400 . . . . . . . . . 10  |-  ( ( ( b  e.  V  /\  -.  b  e.  A
)  /\  a  e.  A )  ->  b  e.  ( V  \  {
a } ) )
3029ex 432 . . . . . . . . 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 428 . . . . . . 7  |-  ( ( a  e.  A  /\  b  e.  B )  ->  b  e.  ( V 
\  { a } ) )
33 neleq1 2720 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  e/  ( <. V ,  E >. Neighbors  a )  <-> 
b  e/  ( <. V ,  E >. Neighbors  a ) ) )
34 fveq2 5774 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( V VDeg  E ) `  y )  =  ( ( V VDeg  E ) `
 b ) )
3534eqeq2d 2396 . . . . . . . . 9  |-  ( y  =  b  ->  (
( ( V VDeg  E
) `  a )  =  ( ( V VDeg 
E ) `  y
)  <->  ( ( V VDeg 
E ) `  a
)  =  ( ( V VDeg  E ) `  b ) ) )
3633, 35imbi12d 318 . . . . . . . 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 3131 . . . . . . 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 2727 . . . . . . . . 9  |-  ( -.  b  e/  ( <. V ,  E >. Neighbors  a
)  <->  b  e.  (
<. V ,  E >. Neighbors  a
) )
40 frisusgra 25113 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  V USGrph  E )
41 nbgraeledg 24551 . . . . . . . . . . . . . . 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 4022 . . . . . . . . . . . . . . 15  |-  { b ,  a }  =  { a ,  b }
4443eleq1i 2459 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 433 . . . . . . . . . 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 eqneqall 2589 . . . . . . . . . 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 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 ) ) ) )
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 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
) ) ) )
5756com3l 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 ) ) ) )
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 2802 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577    e/ wnel 2578   A.wral 2732   {crab 2736    \ cdif 3386   {csn 3944   {cpr 3946   <.cop 3950   class class class wbr 4367   ran crn 4914   ` cfv 5496  (class class class)co 6196   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:  frgrawopreglem5  25169  frgrawopreg1  25171  frgrawopreg2  25172
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