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Theorem frgrancvvdeqlemC 25846
Description: Lemma C for frgrancvvdeq 25849. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-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
frgrancvvdeqlemC  |-  ( ph  ->  A : D -onto-> N
)
Distinct variable groups:    y, D, x    x, V, y    x, E, y    y, Y    ph, y    y, N    x, D    x, N    ph, x
Allowed substitution hints:    A( x, y)    X( x, y)    Y( x)

Proof of Theorem frgrancvvdeqlemC
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.nx . . 3  |-  D  =  ( <. V ,  E >. Neighbors  X )
2 frgrancvvdeq.ny . . 3  |-  N  =  ( <. V ,  E >. Neighbors  Y )
3 frgrancvvdeq.x . . 3  |-  ( ph  ->  X  e.  V )
4 frgrancvvdeq.y . . 3  |-  ( ph  ->  Y  e.  V )
5 frgrancvvdeq.ne . . 3  |-  ( ph  ->  X  =/=  Y )
6 frgrancvvdeq.xy . . 3  |-  ( ph  ->  Y  e/  D )
7 frgrancvvdeq.f . . 3  |-  ( ph  ->  V FriendGrph  E )
8 frgrancvvdeq.a . . 3  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem5 25841 . 2  |-  ( ph  ->  A : D --> N )
107adantr 472 . . . . . . 7  |-  ( (
ph  /\  n  e.  N )  ->  V FriendGrph  E )
112eleq2i 2541 . . . . . . . . . 10  |-  ( n  e.  N  <->  n  e.  ( <. V ,  E >. Neighbors  Y ) )
12 frisusgra 25799 . . . . . . . . . . 11  |-  ( V FriendGrph  E  ->  V USGrph  E )
13 nbgraisvtx 25238 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  V ) )
147, 12, 133syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( n  e.  (
<. V ,  E >. Neighbors  Y
)  ->  n  e.  V ) )
1511, 14syl5bi 225 . . . . . . . . 9  |-  ( ph  ->  ( n  e.  N  ->  n  e.  V ) )
1615imp 436 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  n  e.  V )
173adantr 472 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  X  e.  V )
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 25838 . . . . . . . . . 10  |-  ( ph  ->  X  e/  N )
19 df-nel 2644 . . . . . . . . . . 11  |-  ( X  e/  N  <->  -.  X  e.  N )
20 eleq1 2537 . . . . . . . . . . . . . 14  |-  ( n  =  X  ->  (
n  e.  N  <->  X  e.  N ) )
2120biimpcd 232 . . . . . . . . . . . . 13  |-  ( n  e.  N  ->  (
n  =  X  ->  X  e.  N )
)
2221con3rr3 143 . . . . . . . . . . . 12  |-  ( -.  X  e.  N  -> 
( n  e.  N  ->  -.  n  =  X ) )
23 df-ne 2643 . . . . . . . . . . . 12  |-  ( n  =/=  X  <->  -.  n  =  X )
2422, 23syl6ibr 235 . . . . . . . . . . 11  |-  ( -.  X  e.  N  -> 
( n  e.  N  ->  n  =/=  X ) )
2519, 24sylbi 200 . . . . . . . . . 10  |-  ( X  e/  N  ->  (
n  e.  N  ->  n  =/=  X ) )
2618, 25syl 17 . . . . . . . . 9  |-  ( ph  ->  ( n  e.  N  ->  n  =/=  X ) )
2726imp 436 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  n  =/=  X )
2816, 17, 273jca 1210 . . . . . . 7  |-  ( (
ph  /\  n  e.  N )  ->  (
n  e.  V  /\  X  e.  V  /\  n  =/=  X ) )
2910, 28jca 541 . . . . . 6  |-  ( (
ph  /\  n  e.  N )  ->  ( V FriendGrph  E  /\  ( n  e.  V  /\  X  e.  V  /\  n  =/=  X ) ) )
30 frgraun 25803 . . . . . . 7  |-  ( V FriendGrph  E  ->  ( ( n  e.  V  /\  X  e.  V  /\  n  =/=  X )  ->  E! m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )
3130imp 436 . . . . . 6  |-  ( ( V FriendGrph  E  /\  (
n  e.  V  /\  X  e.  V  /\  n  =/=  X ) )  ->  E! m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )
32 reurex 2995 . . . . . . 7  |-  ( E! m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E )  ->  E. m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )
33 df-rex 2762 . . . . . . 7  |-  ( E. m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E )  <->  E. m
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )
3432, 33sylib 201 . . . . . 6  |-  ( E! m  e.  V  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E )  ->  E. m ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  {
m ,  X }  e.  ran  E ) ) )
3529, 31, 343syl 18 . . . . 5  |-  ( (
ph  /\  n  e.  N )  ->  E. m
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )
367, 12syl 17 . . . . . . . 8  |-  ( ph  ->  V USGrph  E )
37 simprrr 783 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  { m ,  X }  e.  ran  E )
381eleq2i 2541 . . . . . . . . . . . . 13  |-  ( m  e.  D  <->  m  e.  ( <. V ,  E >. Neighbors  X ) )
39 nbgraeledg 25237 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( m  e.  ( <. V ,  E >. Neighbors  X )  <->  { m ,  X }  e.  ran  E ) )
4039ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  ->  (
m  e.  ( <. V ,  E >. Neighbors  X
)  <->  { m ,  X }  e.  ran  E ) )
4140adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  ( m  e.  ( <. V ,  E >. Neighbors  X )  <->  { m ,  X }  e.  ran  E ) )
4238, 41syl5bb 265 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  ( m  e.  D  <->  { m ,  X }  e.  ran  E ) )
4337, 42mpbird 240 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  m  e.  D )
44 nbgraeledg 25237 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  m )  <->  { n ,  m }  e.  ran  E ) )
4544biimprcd 233 . . . . . . . . . . . . . . . . 17  |-  ( { n ,  m }  e.  ran  E  ->  ( V USGrph  E  ->  n  e.  ( <. V ,  E >. Neighbors  m ) ) )
4645adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E )  -> 
( V USGrph  E  ->  n  e.  ( <. V ,  E >. Neighbors  m ) ) )
4746adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  ( V USGrph  E  ->  n  e.  ( <. V ,  E >. Neighbors  m
) ) )
4847com12 31 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  n  e.  (
<. V ,  E >. Neighbors  m
) ) )
4948ad2antlr 741 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  ->  (
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  n  e.  ( <. V ,  E >. Neighbors  m ) ) )
5049imp 436 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  n  e.  ( <. V ,  E >. Neighbors  m ) )
51 elin 3608 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ( <. V ,  E >. Neighbors  m
)  i^i  N )  <->  ( n  e.  ( <. V ,  E >. Neighbors  m
)  /\  n  e.  N ) )
52 simpll 768 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ph )
5339bicomd 206 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( V USGrph  E  ->  ( { m ,  X }  e.  ran  E  <-> 
m  e.  ( <. V ,  E >. Neighbors  X
) ) )
5453adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  V USGrph  E )  ->  ( { m ,  X }  e.  ran  E  <-> 
m  e.  ( <. V ,  E >. Neighbors  X
) ) )
5554biimpa 492 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  m  e.  ( <. V ,  E >. Neighbors  X ) )
5655, 38sylibr 217 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  m  e.  D )
5752, 56jca 541 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ( ph  /\  m  e.  D ) )
58 preq1 4042 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  m  ->  { x ,  y }  =  { m ,  y } )
5958eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  m  ->  ( { x ,  y }  e.  ran  E  <->  { m ,  y }  e.  ran  E ) )
6059riotabidv 6272 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  m  ->  ( iota_ y  e.  N  {
x ,  y }  e.  ran  E )  =  ( iota_ y  e.  N  { m ,  y }  e.  ran  E ) )
6160cbvmptv 4488 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )  =  ( m  e.  D  |->  ( iota_ y  e.  N  { m ,  y }  e.  ran  E
) )
628, 61eqtri 2493 . . . . . . . . . . . . . . . . . . . . 21  |-  A  =  ( m  e.  D  |->  ( iota_ y  e.  N  { m ,  y }  e.  ran  E
) )
631, 2, 3, 4, 5, 6, 7, 62frgrancvvdeqlem6 25842 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  m  e.  D )  ->  { ( A `  m ) }  =  ( (
<. V ,  E >. Neighbors  m
)  i^i  N )
)
64 eleq2 2538 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  =  { ( A `  m ) }  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  <->  n  e.  { ( A `  m ) } ) )
6564eqcoms 2479 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { ( A `  m
) }  =  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  <->  n  e.  { ( A `  m ) } ) )
66 elsni 3985 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  { ( A `
 m ) }  ->  n  =  ( A `  m ) )
6765, 66syl6bi 236 . . . . . . . . . . . . . . . . . . . 20  |-  ( { ( A `  m
) }  =  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) )
6857, 63, 673syl 18 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) )
6968expcom 442 . . . . . . . . . . . . . . . . . 18  |-  ( { m ,  X }  e.  ran  E  ->  (
( ph  /\  V USGrph  E
)  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) ) )
7069ad2antll 743 . . . . . . . . . . . . . . . . 17  |-  ( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  ( ( ph  /\  V USGrph  E )  -> 
( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) ) )
7170com3r 81 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ( <. V ,  E >. Neighbors  m
)  i^i  N )  ->  ( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  {
m ,  X }  e.  ran  E ) )  ->  ( ( ph  /\  V USGrph  E )  ->  n  =  ( A `  m ) ) ) )
7251, 71sylbir 218 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( <. V ,  E >. Neighbors  m
)  /\  n  e.  N )  ->  (
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  (
( ph  /\  V USGrph  E
)  ->  n  =  ( A `  m ) ) ) )
7372ex 441 . . . . . . . . . . . . . 14  |-  ( n  e.  ( <. V ,  E >. Neighbors  m )  ->  (
n  e.  N  -> 
( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  {
m ,  X }  e.  ran  E ) )  ->  ( ( ph  /\  V USGrph  E )  ->  n  =  ( A `  m ) ) ) ) )
7473com14 90 . . . . . . . . . . . . 13  |-  ( (
ph  /\  V USGrph  E )  ->  ( n  e.  N  ->  ( (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  ( n  e.  ( <. V ,  E >. Neighbors  m )  ->  n  =  ( A `  m ) ) ) ) )
7574imp31 439 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  ( n  e.  ( <. V ,  E >. Neighbors  m )  ->  n  =  ( A `  m ) ) )
7650, 75mpd 15 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  n  =  ( A `  m ) )
7743, 76jca 541 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  /\  (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )  ->  ( m  e.  D  /\  n  =  ( A `  m ) ) )
7877ex 441 . . . . . . . . 9  |-  ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  ->  (
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  (
m  e.  D  /\  n  =  ( A `  m ) ) ) )
7978ex 441 . . . . . . . 8  |-  ( (
ph  /\  V USGrph  E )  ->  ( n  e.  N  ->  ( (
m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  ( m  e.  D  /\  n  =  ( A `  m
) ) ) ) )
8036, 79mpdan 681 . . . . . . 7  |-  ( ph  ->  ( n  e.  N  ->  ( ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  {
m ,  X }  e.  ran  E ) )  ->  ( m  e.  D  /\  n  =  ( A `  m
) ) ) ) )
8180imp 436 . . . . . 6  |-  ( (
ph  /\  n  e.  N )  ->  (
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) )  ->  (
m  e.  D  /\  n  =  ( A `  m ) ) ) )
8281eximdv 1772 . . . . 5  |-  ( (
ph  /\  n  e.  N )  ->  ( E. m ( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  {
m ,  X }  e.  ran  E ) )  ->  E. m ( m  e.  D  /\  n  =  ( A `  m ) ) ) )
8335, 82mpd 15 . . . 4  |-  ( (
ph  /\  n  e.  N )  ->  E. m
( m  e.  D  /\  n  =  ( A `  m )
) )
84 df-rex 2762 . . . 4  |-  ( E. m  e.  D  n  =  ( A `  m )  <->  E. m
( m  e.  D  /\  n  =  ( A `  m )
) )
8583, 84sylibr 217 . . 3  |-  ( (
ph  /\  n  e.  N )  ->  E. m  e.  D  n  =  ( A `  m ) )
8685ralrimiva 2809 . 2  |-  ( ph  ->  A. n  e.  N  E. m  e.  D  n  =  ( A `  m ) )
87 dffo3 6052 . 2  |-  ( A : D -onto-> N  <->  ( A : D --> N  /\  A. n  e.  N  E. m  e.  D  n  =  ( A `  m ) ) )
889, 86, 87sylanbrc 677 1  |-  ( ph  ->  A : D -onto-> N
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641    e/ wnel 2642   A.wral 2756   E.wrex 2757   E!wreu 2758    i^i cin 3389   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395    |-> cmpt 4454   ran crn 4840   -->wf 5585   -onto->wfo 5587   ` cfv 5589   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-rmo 2764  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:  frgrancvvdeqlem8  25847
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