MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrancvvdeqlem4 Structured version   Visualization version   Unicode version

Theorem frgrancvvdeqlem4 25840
Description: Lemma 4 for frgrancvvdeq 25849. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx  |-  D  =  ( <. V ,  E >. Neighbors  X )
frgrancvvdeq.ny  |-  N  =  ( <. V ,  E >. Neighbors  Y )
frgrancvvdeq.x  |-  ( ph  ->  X  e.  V )
frgrancvvdeq.y  |-  ( ph  ->  Y  e.  V )
frgrancvvdeq.ne  |-  ( ph  ->  X  =/=  Y )
frgrancvvdeq.xy  |-  ( ph  ->  Y  e/  D )
frgrancvvdeq.f  |-  ( ph  ->  V FriendGrph  E )
frgrancvvdeq.a  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
Assertion
Ref Expression
frgrancvvdeqlem4  |-  ( (
ph  /\  x  e.  D )  ->  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N
) )
Distinct variable groups:    y, D    x, y, V    x, E, y    y, Y    ph, y    y, N
Allowed substitution hints:    ph( x)    A( x, y)    D( x)    N( x)    X( x, y)    Y( x)

Proof of Theorem frgrancvvdeqlem4
Dummy variables  a 
b  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.ny . . 3  |-  N  =  ( <. V ,  E >. Neighbors  Y )
21ineq2i 3622 . 2  |-  ( (
<. V ,  E >. Neighbors  x
)  i^i  N )  =  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )
3 frgrancvvdeq.nx . . . . . . 7  |-  D  =  ( <. V ,  E >. Neighbors  X )
43eleq2i 2541 . . . . . 6  |-  ( x  e.  D  <->  x  e.  ( <. V ,  E >. Neighbors  X ) )
5 frgrancvvdeq.f . . . . . . . 8  |-  ( ph  ->  V FriendGrph  E )
6 frisusgra 25799 . . . . . . . 8  |-  ( V FriendGrph  E  ->  V USGrph  E )
75, 6syl 17 . . . . . . 7  |-  ( ph  ->  V USGrph  E )
8 nbgraisvtx 25238 . . . . . . 7  |-  ( V USGrph  E  ->  ( x  e.  ( <. V ,  E >. Neighbors  X )  ->  x  e.  V ) )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  ( x  e.  (
<. V ,  E >. Neighbors  X
)  ->  x  e.  V ) )
104, 9syl5bi 225 . . . . 5  |-  ( ph  ->  ( x  e.  D  ->  x  e.  V ) )
1110imp 436 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  V )
12 frgrancvvdeq.x . . . . 5  |-  ( ph  ->  X  e.  V )
13 frgrancvvdeq.y . . . . 5  |-  ( ph  ->  Y  e.  V )
14 frgrancvvdeq.ne . . . . 5  |-  ( ph  ->  X  =/=  Y )
15 frgrancvvdeq.xy . . . . 5  |-  ( ph  ->  Y  e/  D )
16 frgrancvvdeq.a . . . . 5  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
173, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem1 25837 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
185adantr 472 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  V FriendGrph  E )
19 frisusgranb 25804 . . . . 5  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } )
2018, 19syl 17 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  A. a  e.  V  A. b  e.  ( V  \  {
a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors 
a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )
2111, 17, 20jca31 543 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( x  e.  V  /\  Y  e.  ( V  \  { x }
) )  /\  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } ) )
22 sneq 3969 . . . . . . . . 9  |-  ( a  =  x  ->  { a }  =  { x } )
2322difeq2d 3540 . . . . . . . 8  |-  ( a  =  x  ->  ( V  \  { a } )  =  ( V 
\  { x }
) )
24 oveq2 6316 . . . . . . . . . . 11  |-  ( a  =  x  ->  ( <. V ,  E >. Neighbors  a
)  =  ( <. V ,  E >. Neighbors  x
) )
2524ineq1d 3624 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( <. V ,  E >. Neighbors 
a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) ) )
2625eqeq1d 2473 . . . . . . . . 9  |-  ( a  =  x  ->  (
( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  <->  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } ) )
2726rexbidv 2892 . . . . . . . 8  |-  ( a  =  x  ->  ( E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  <->  E. n  e.  V  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } ) )
2823, 27raleqbidv 2987 . . . . . . 7  |-  ( a  =  x  ->  ( A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } 
<-> 
A. b  e.  ( V  \  { x } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } ) )
2928rspcva 3134 . . . . . 6  |-  ( ( x  e.  V  /\  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )  ->  A. b  e.  ( V  \  { x }
) E. n  e.  V  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  b ) )  =  { n } )
30 oveq2 6316 . . . . . . . . . . . 12  |-  ( b  =  Y  ->  ( <. V ,  E >. Neighbors  b
)  =  ( <. V ,  E >. Neighbors  Y
) )
3130ineq2d 3625 . . . . . . . . . . 11  |-  ( b  =  Y  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) ) )
3231eqeq1d 2473 . . . . . . . . . 10  |-  ( b  =  Y  ->  (
( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  <->  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )  =  { n } ) )
3332rexbidv 2892 . . . . . . . . 9  |-  ( b  =  Y  ->  ( E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  <->  E. n  e.  V  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )  =  { n } ) )
3433rspcva 3134 . . . . . . . 8  |-  ( ( Y  e.  ( V 
\  { x }
)  /\  A. b  e.  ( V  \  {
x } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )  ->  E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n } )
35 ssnid 3989 . . . . . . . . . . . . . . 15  |-  n  e. 
{ n }
36 eleq2 2538 . . . . . . . . . . . . . . . . 17  |-  ( { n }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  ->  (
n  e.  { n } 
<->  n  e.  ( (
<. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) ) ) )
3736eqcoms 2479 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
n  e.  { n } 
<->  n  e.  ( (
<. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) ) ) )
38 elin 3608 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )  <->  ( n  e.  ( <. V ,  E >. Neighbors  x )  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) ) )
3938biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) ) )
4037, 39syl6bi 236 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
n  e.  { n }  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) ) ) )
4135, 40mpi 20 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) ) )
42 nbgraeledg 25237 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { n ,  x }  e.  ran  E ) )
43 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { n ,  x }  =  {
x ,  n }
4443eleq1i 2540 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { n ,  x }  e.  ran  E  <->  { x ,  n }  e.  ran  E )
4542, 44syl6bb 269 . . . . . . . . . . . . . . . . . . . . 21  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { x ,  n }  e.  ran  E ) )
4645biimpd 212 . . . . . . . . . . . . . . . . . . . 20  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  ->  { x ,  n }  e.  ran  E ) )
477, 46syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( n  e.  (
<. V ,  E >. Neighbors  x
)  ->  { x ,  n }  e.  ran  E ) )
4847adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  D )  ->  (
n  e.  ( <. V ,  E >. Neighbors  x
)  ->  { x ,  n }  e.  ran  E ) )
4948com12 31 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  x )  ->  (
( ph  /\  x  e.  D )  ->  { x ,  n }  e.  ran  E ) )
5049adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  -> 
( ( ph  /\  x  e.  D )  ->  { x ,  n }  e.  ran  E ) )
5150imp 436 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  (
<. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  /\  ( ph  /\  x  e.  D ) )  ->  { x ,  n }  e.  ran  E )
521eqcomi 2480 . . . . . . . . . . . . . . . . . . 19  |-  ( <. V ,  E >. Neighbors  Y
)  =  N
5352eleq2i 2541 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  <->  n  e.  N )
5453biimpi 199 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  N )
5554adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  ->  n  e.  N )
563, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem3 25839 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
57 preq2 4043 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  n  ->  { x ,  y }  =  { x ,  n } )
5857eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( y  =  n  ->  ( { x ,  y }  e.  ran  E  <->  { x ,  n }  e.  ran  E ) )
5958riota2 6292 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  N  /\  E! y  e.  N  { x ,  y }  e.  ran  E
)  ->  ( {
x ,  n }  e.  ran  E  <->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E )  =  n ) )
6055, 56, 59syl2an 485 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  (
<. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  /\  ( ph  /\  x  e.  D ) )  -> 
( { x ,  n }  e.  ran  E  <-> 
( iota_ y  e.  N  { x ,  y }  e.  ran  E
)  =  n ) )
6151, 60mpbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( n  e.  (
<. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  /\  ( ph  /\  x  e.  D ) )  -> 
( iota_ y  e.  N  { x ,  y }  e.  ran  E
)  =  n )
6241, 61sylan 479 . . . . . . . . . . . . 13  |-  ( ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  /\  ( ph  /\  x  e.  D
) )  ->  ( iota_ y  e.  N  {
x ,  y }  e.  ran  E )  =  n )
6362eqcomd 2477 . . . . . . . . . . . 12  |-  ( ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  /\  ( ph  /\  x  e.  D
) )  ->  n  =  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )
6463sneqd 3971 . . . . . . . . . . 11  |-  ( ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  /\  ( ph  /\  x  e.  D
) )  ->  { n }  =  { ( iota_ y  e.  N  {
x ,  y }  e.  ran  E ) } )
65 eqeq1 2475 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  <->  { n }  =  { ( iota_ y  e.  N  {
x ,  y }  e.  ran  E ) } ) )
6665adantr 472 . . . . . . . . . . 11  |-  ( ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  /\  ( ph  /\  x  e.  D
) )  ->  (
( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  <->  { n }  =  { ( iota_ y  e.  N  {
x ,  y }  e.  ran  E ) } ) )
6764, 66mpbird 240 . . . . . . . . . 10  |-  ( ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  /\  ( ph  /\  x  e.  D
) )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } )
6867ex 441 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
( ph  /\  x  e.  D )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) )
6968rexlimivw 2869 . . . . . . . 8  |-  ( E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
n }  ->  (
( ph  /\  x  e.  D )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) )
7034, 69syl 17 . . . . . . 7  |-  ( ( Y  e.  ( V 
\  { x }
)  /\  A. b  e.  ( V  \  {
x } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )  -> 
( ( ph  /\  x  e.  D )  ->  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) )
7170expcom 442 . . . . . 6  |-  ( A. b  e.  ( V  \  { x } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  ->  ( Y  e.  ( V  \  { x } )  ->  ( ( ph  /\  x  e.  D )  ->  ( ( <. V ,  E >. Neighbors  x
)  i^i  ( <. V ,  E >. Neighbors  Y ) )  =  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) ) )
7229, 71syl 17 . . . . 5  |-  ( ( x  e.  V  /\  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )  -> 
( Y  e.  ( V  \  { x } )  ->  (
( ph  /\  x  e.  D )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) ) )
7372impancom 447 . . . 4  |-  ( ( x  e.  V  /\  Y  e.  ( V  \  { x } ) )  ->  ( A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n }  ->  (
( ph  /\  x  e.  D )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) ) )
7473imp 436 . . 3  |-  ( ( ( x  e.  V  /\  Y  e.  ( V  \  { x }
) )  /\  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. n  e.  V  ( ( <. V ,  E >. Neighbors  a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  {
n } )  -> 
( ( ph  /\  x  e.  D )  ->  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } ) )
7521, 74mpcom 36 . 2  |-  ( (
ph  /\  x  e.  D )  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) )  =  {
( iota_ y  e.  N  { x ,  y }  e.  ran  E
) } )
762, 75syl5req 2518 1  |-  ( (
ph  /\  x  e.  D )  ->  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    e/ wnel 2642   A.wral 2756   E.wrex 2757   E!wreu 2758    \ cdif 3387    i^i cin 3389   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395    |-> cmpt 4454   ran crn 4840   iota_crio 6269  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224   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-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-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-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  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-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-frgra 25796
This theorem is referenced by:  frgrancvvdeqlem6  25842
  Copyright terms: Public domain W3C validator