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Theorem frgrancvvdeqlem4 24857
Description: Lemma 4 for frgrancvvdeq 24866. 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 3702 . 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 2545 . . . . . 6  |-  ( x  e.  D  <->  x  e.  ( <. V ,  E >. Neighbors  X ) )
5 frgrancvvdeq.f . . . . . . . 8  |-  ( ph  ->  V FriendGrph  E )
6 frisusgra 24815 . . . . . . . 8  |-  ( V FriendGrph  E  ->  V USGrph  E )
75, 6syl 16 . . . . . . 7  |-  ( ph  ->  V USGrph  E )
8 nbgraisvtx 24254 . . . . . . 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 24854 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
185adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  V FriendGrph  E )
19 frisusgranb 24820 . . . . 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 534 . . 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 4043 . . . . . . . . 9  |-  ( a  =  x  ->  { a }  =  { x } )
2322difeq2d 3627 . . . . . . . 8  |-  ( a  =  x  ->  ( V  \  { a } )  =  ( V 
\  { x }
) )
24 oveq2 6303 . . . . . . . . . . 11  |-  ( a  =  x  ->  ( <. V ,  E >. Neighbors  a
)  =  ( <. V ,  E >. Neighbors  x
) )
2524ineq1d 3704 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( <. V ,  E >. Neighbors 
a )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) ) )
2625eqeq1d 2469 . . . . . . . . 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 2978 . . . . . . . 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 3077 . . . . . . 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 3217 . . . . . 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 6303 . . . . . . . . . . . 12  |-  ( b  =  Y  ->  ( <. V ,  E >. Neighbors  b
)  =  ( <. V ,  E >. Neighbors  Y
) )
3130ineq2d 3705 . . . . . . . . . . 11  |-  ( b  =  Y  ->  (
( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  b
) )  =  ( ( <. V ,  E >. Neighbors  x )  i^i  ( <. V ,  E >. Neighbors  Y
) ) )
3231eqeq1d 2469 . . . . . . . . . 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 2978 . . . . . . . . 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 3217 . . . . . . . 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 4062 . . . . . . . . . . . . . . 15  |-  n  e. 
{ n }
36 eleq2 2540 . . . . . . . . . . . . . . . . 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 3692 . . . . . . . . . . . . . . . . 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 24253 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  x )  <->  { n ,  x }  e.  ran  E ) )
43 prcom 4111 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { n ,  x }  =  {
x ,  n }
4443eleq1i 2544 . . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 2480 . . . . . . . . . . . . . . . . . . 19  |-  ( <. V ,  E >. Neighbors  Y
)  =  N
5352eleq2i 2545 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  <->  n  e.  N )
5453biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  N )
5554adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  x
)  /\  n  e.  ( <. V ,  E >. Neighbors  Y ) )  ->  n  e.  N )
563, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem3 24856 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  D )  ->  E! y  e.  N  {
x ,  y }  e.  ran  E )
57 preq2 4113 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  n  ->  { x ,  y }  =  { x ,  n } )
5857eleq1d 2536 . . . . . . . . . . . . . . . . 17  |-  ( y  =  n  ->  ( { x ,  y }  e.  ran  E  <->  { x ,  n }  e.  ran  E ) )
5958riota2 6279 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 4045 . . . . . . . . . . 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 2471 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 2956 . . . . . . . 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 440 . . . 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 2521 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 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2817   E.wrex 2818   E!wreu 2819    \ cdif 3478    i^i cin 3480   {csn 4033   {cpr 4035   <.cop 4039   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   iota_crio 6255  (class class class)co 6295   USGrph cusg 24153   Neighbors cnbgra 24240   FriendGrph cfrgra 24811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-frgra 24812
This theorem is referenced by:  frgrancvvdeqlem6  24859
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