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Theorem frgrancvvdeqlem4 25759
Description: Lemma 4 for frgrancvvdeq 25768. 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 3661 . 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 2499 . . . . . 6  |-  ( x  e.  D  <->  x  e.  ( <. V ,  E >. Neighbors  X ) )
5 frgrancvvdeq.f . . . . . . . 8  |-  ( ph  ->  V FriendGrph  E )
6 frisusgra 25718 . . . . . . . 8  |-  ( V FriendGrph  E  ->  V USGrph  E )
75, 6syl 17 . . . . . . 7  |-  ( ph  ->  V USGrph  E )
8 nbgraisvtx 25157 . . . . . . 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 220 . . . . 5  |-  ( ph  ->  ( x  e.  D  ->  x  e.  V ) )
1110imp 430 . . . 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 25756 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
185adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  V FriendGrph  E )
19 frisusgranb 25723 . . . . 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 536 . . 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 4008 . . . . . . . . 9  |-  ( a  =  x  ->  { a }  =  { x } )
2322difeq2d 3583 . . . . . . . 8  |-  ( a  =  x  ->  ( V  \  { a } )  =  ( V 
\  { x }
) )
24 oveq2 6313 . . . . . . . . . . 11  |-  ( a  =  x  ->  ( <. V ,  E >. Neighbors  a
)  =  ( <. V ,  E >. Neighbors  x
) )
2524ineq1d 3663 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( <. V ,  E >. Neighbors 
a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) ) )
2625eqeq1d 2424 . . . . . . . . 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 2936 . . . . . . . 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 3036 . . . . . . 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 3180 . . . . . 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 6313 . . . . . . . . . . . 12  |-  ( b  =  Y  ->  ( <. V ,  E >. Neighbors  b
)  =  ( <. V ,  E >. Neighbors  Y
) )
3130ineq2d 3664 . . . . . . . . . . 11  |-  ( b  =  Y  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) ) )
3231eqeq1d 2424 . . . . . . . . . 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 2936 . . . . . . . . 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 3180 . . . . . . . 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 4027 . . . . . . . . . . . . . . 15  |-  n  e. 
{ n }
36 eleq2 2496 . . . . . . . . . . . . . . . . 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 2434 . . . . . . . . . . . . . . . 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 3649 . . . . . . . . . . . . . . . . 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 197 . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . 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 25156 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { n ,  x }  e.  ran  E ) )
43 prcom 4078 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { n ,  x }  =  {
x ,  n }
4443eleq1i 2498 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { n ,  x }  e.  ran  E  <->  { x ,  n }  e.  ran  E )
4542, 44syl6bb 264 . . . . . . . . . . . . . . . . . . . . 21  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { x ,  n }  e.  ran  E ) )
4645biimpd 210 . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  D )  ->  (
n  e.  ( <. V ,  E >. Neighbors  x
)  ->  { x ,  n }  e.  ran  E ) )
4948com12 32 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  x )  ->  (
( ph  /\  x  e.  D )  ->  { x ,  n }  e.  ran  E ) )
5049adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  -> 
( ( ph  /\  x  e.  D )  ->  { x ,  n }  e.  ran  E ) )
5150imp 430 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  (
<. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  /\  ( ph  /\  x  e.  D ) )  ->  { x ,  n }  e.  ran  E )
521eqcomi 2435 . . . . . . . . . . . . . . . . . . 19  |-  ( <. V ,  E >. Neighbors  Y
)  =  N
5352eleq2i 2499 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  <->  n  e.  N )
5453biimpi 197 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  N )
5554adantl 467 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  ->  n  e.  N )
563, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem3 25758 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
57 preq2 4080 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  n  ->  { x ,  y }  =  { x ,  n } )
5857eleq1d 2491 . . . . . . . . . . . . . . . . 17  |-  ( y  =  n  ->  ( { x ,  y }  e.  ran  E  <->  { x ,  n }  e.  ran  E ) )
5958riota2 6289 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 2430 . . . . . . . . . . . 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 4010 . . . . . . . . . . 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 2426 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 235 . . . . . . . . . 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 435 . . . . . . . . 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 2911 . . . . . . . 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 436 . . . . . 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 441 . . . 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 430 . . 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 37 . 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 2476 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614    e/ wnel 2615   A.wral 2771   E.wrex 2772   E!wreu 2773    \ cdif 3433    i^i cin 3435   {csn 3998   {cpr 4000   <.cop 4004   class class class wbr 4423    |-> cmpt 4482   ran crn 4854   iota_crio 6266  (class class class)co 6305   USGrph cusg 25055   Neighbors cnbgra 25143   FriendGrph cfrgra 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522  df-usgra 25058  df-nbgra 25146  df-frgra 25715
This theorem is referenced by:  frgrancvvdeqlem6  25761
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