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Theorem frgrancvvdeqlem4 30535
Description: Lemma 4 for frgrancvvdeq 30544. 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 3546 . 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 2505 . . . . . 6  |-  ( x  e.  D  <->  x  e.  ( <. V ,  E >. Neighbors  X ) )
5 frgrancvvdeq.f . . . . . . . 8  |-  ( ph  ->  V FriendGrph  E )
6 frisusgra 30493 . . . . . . . 8  |-  ( V FriendGrph  E  ->  V USGrph  E )
75, 6syl 16 . . . . . . 7  |-  ( ph  ->  V USGrph  E )
8 nbgraisvtx 23261 . . . . . . 7  |-  ( V USGrph  E  ->  ( x  e.  ( <. V ,  E >. Neighbors  X )  ->  x  e.  V ) )
97, 8syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  (
<. V ,  E >. Neighbors  X
)  ->  x  e.  V ) )
104, 9syl5bi 217 . . . . 5  |-  ( ph  ->  ( x  e.  D  ->  x  e.  V ) )
1110imp 429 . . . 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 30532 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
185adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  V FriendGrph  E )
19 frisusgranb 30498 . . . . 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 16 . . . 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 531 . . 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 3884 . . . . . . . . 9  |-  ( a  =  x  ->  { a }  =  { x } )
2322difeq2d 3471 . . . . . . . 8  |-  ( a  =  x  ->  ( V  \  { a } )  =  ( V 
\  { x }
) )
24 oveq2 6098 . . . . . . . . . . 11  |-  ( a  =  x  ->  ( <. V ,  E >. Neighbors  a
)  =  ( <. V ,  E >. Neighbors  x
) )
2524ineq1d 3548 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( <. V ,  E >. Neighbors 
a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) ) )
2625eqeq1d 2449 . . . . . . . . 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 2734 . . . . . . . 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 2929 . . . . . . 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 3068 . . . . . 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 6098 . . . . . . . . . . . 12  |-  ( b  =  Y  ->  ( <. V ,  E >. Neighbors  b
)  =  ( <. V ,  E >. Neighbors  Y
) )
3130ineq2d 3549 . . . . . . . . . . 11  |-  ( b  =  Y  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) ) )
3231eqeq1d 2449 . . . . . . . . . 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 2734 . . . . . . . . 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 3068 . . . . . . . 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 3903 . . . . . . . . . . . . . . 15  |-  n  e. 
{ n }
36 eleq2 2502 . . . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . . 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 3536 . . . . . . . . . . . . . . . . 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 194 . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . 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 23260 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { n ,  x }  e.  ran  E ) )
43 prcom 3950 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { n ,  x }  =  {
x ,  n }
4443eleq1i 2504 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { n ,  x }  e.  ran  E  <->  { x ,  n }  e.  ran  E )
4542, 44syl6bb 261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { x ,  n }  e.  ran  E ) )
4645biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  ->  { x ,  n }  e.  ran  E ) )
477, 46syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( n  e.  (
<. V ,  E >. Neighbors  x
)  ->  { x ,  n }  e.  ran  E ) )
4847adantr 462 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  -> 
( ( ph  /\  x  e.  D )  ->  { x ,  n }  e.  ran  E ) )
5150imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( n  e.  (
<. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  /\  ( ph  /\  x  e.  D ) )  ->  { x ,  n }  e.  ran  E )
521eqcomi 2445 . . . . . . . . . . . . . . . . . . 19  |-  ( <. V ,  E >. Neighbors  Y
)  =  N
5352eleq2i 2505 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  <->  n  e.  N )
5453biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  N )
5554adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  ->  n  e.  N )
563, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem3 30534 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
57 preq2 3952 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  n  ->  { x ,  y }  =  { x ,  n } )
5857eleq1d 2507 . . . . . . . . . . . . . . . . 17  |-  ( y  =  n  ->  ( { x ,  y }  e.  ran  E  <->  { x ,  n }  e.  ran  E ) )
5958riota2 6073 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 2446 . . . . . . . . . . . 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 3886 . . . . . . . . . . 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 2447 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 232 . . . . . . . . . 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 434 . . . . . . . . 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 2835 . . . . . . . 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 16 . . . . . . 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 435 . . . . . 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 16 . . . . 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 438 . . . 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 429 . . 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 2486 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 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604    e/ wnel 2605   A.wral 2713   E.wrex 2714   E!wreu 2715    \ cdif 3322    i^i cin 3324   {csn 3874   {cpr 3876   <.cop 3880   class class class wbr 4289    e. cmpt 4347   ran crn 4837   iota_crio 6048  (class class class)co 6090   USGrph cusg 23183   Neighbors cnbgra 23248   FriendGrph cfrgra 30489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-usgra 23185  df-nbgra 23251  df-frgra 30490
This theorem is referenced by:  frgrancvvdeqlem6  30537
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