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Theorem frgrancvvdeqlemC 30655
Description: Lemma C for frgrancvvdeq 30658. This corresponds to the following observation 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 30650 . 2  |-  ( ph  ->  A : D --> N )
107adantr 465 . . . . . . 7  |-  ( (
ph  /\  n  e.  N )  ->  V FriendGrph  E )
112eleq2i 2507 . . . . . . . . . 10  |-  ( n  e.  N  <->  n  e.  ( <. V ,  E >. Neighbors  Y ) )
12 frisusgra 30607 . . . . . . . . . . 11  |-  ( V FriendGrph  E  ->  V USGrph  E )
13 nbgraisvtx 23361 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  Y )  ->  n  e.  V ) )
147, 12, 133syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( n  e.  (
<. V ,  E >. Neighbors  Y
)  ->  n  e.  V ) )
1511, 14syl5bi 217 . . . . . . . . 9  |-  ( ph  ->  ( n  e.  N  ->  n  e.  V ) )
1615imp 429 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  n  e.  V )
173adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  X  e.  V )
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 30647 . . . . . . . . . 10  |-  ( ph  ->  X  e/  N )
19 df-nel 2623 . . . . . . . . . . 11  |-  ( X  e/  N  <->  -.  X  e.  N )
20 eleq1 2503 . . . . . . . . . . . . . 14  |-  ( n  =  X  ->  (
n  e.  N  <->  X  e.  N ) )
2120biimpcd 224 . . . . . . . . . . . . 13  |-  ( n  e.  N  ->  (
n  =  X  ->  X  e.  N )
)
2221con3rr3 136 . . . . . . . . . . . 12  |-  ( -.  X  e.  N  -> 
( n  e.  N  ->  -.  n  =  X ) )
23 df-ne 2622 . . . . . . . . . . . 12  |-  ( n  =/=  X  <->  -.  n  =  X )
2422, 23syl6ibr 227 . . . . . . . . . . 11  |-  ( -.  X  e.  N  -> 
( n  e.  N  ->  n  =/=  X ) )
2519, 24sylbi 195 . . . . . . . . . 10  |-  ( X  e/  N  ->  (
n  e.  N  ->  n  =/=  X ) )
2618, 25syl 16 . . . . . . . . 9  |-  ( ph  ->  ( n  e.  N  ->  n  =/=  X ) )
2726imp 429 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N )  ->  n  =/=  X )
2816, 17, 273jca 1168 . . . . . . 7  |-  ( (
ph  /\  n  e.  N )  ->  (
n  e.  V  /\  X  e.  V  /\  n  =/=  X ) )
2910, 28jca 532 . . . . . 6  |-  ( (
ph  /\  n  e.  N )  ->  ( V FriendGrph  E  /\  ( n  e.  V  /\  X  e.  V  /\  n  =/=  X ) ) )
30 frgraun 30611 . . . . . . 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 429 . . . . . 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 2956 . . . . . . 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 2740 . . . . . . 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 196 . . . . . 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 20 . . . . 5  |-  ( (
ph  /\  n  e.  N )  ->  E. m
( m  e.  V  /\  ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E ) ) )
367, 12syl 16 . . . . . . . 8  |-  ( ph  ->  V USGrph  E )
37 simprrr 764 . . . . . . . . . . . 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 2507 . . . . . . . . . . . . 13  |-  ( m  e.  D  <->  m  e.  ( <. V ,  E >. Neighbors  X ) )
39 nbgraeledg 23360 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( m  e.  ( <. V ,  E >. Neighbors  X )  <->  { m ,  X }  e.  ran  E ) )
4039ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  V USGrph  E )  /\  n  e.  N )  ->  (
m  e.  ( <. V ,  E >. Neighbors  X
)  <->  { m ,  X }  e.  ran  E ) )
4140adantr 465 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 232 . . . . . . . . . . 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 23360 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( n  e.  ( <. V ,  E >. Neighbors  m )  <->  { n ,  m }  e.  ran  E ) )
4544biimprcd 225 . . . . . . . . . . . . . . . . 17  |-  ( { n ,  m }  e.  ran  E  ->  ( V USGrph  E  ->  n  e.  ( <. V ,  E >. Neighbors  m ) ) )
4645adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( { n ,  m }  e.  ran  E  /\  { m ,  X }  e.  ran  E )  -> 
( V USGrph  E  ->  n  e.  ( <. V ,  E >. Neighbors  m ) ) )
4746adantl 466 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . 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 429 . . . . . . . . . . . 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 3558 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ( <. V ,  E >. Neighbors  m
)  i^i  N )  <->  ( n  e.  ( <. V ,  E >. Neighbors  m
)  /\  n  e.  N ) )
52 simpll 753 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ph )
5339bicomd 201 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( V USGrph  E  ->  ( { m ,  X }  e.  ran  E  <-> 
m  e.  ( <. V ,  E >. Neighbors  X
) ) )
5453adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  V USGrph  E )  ->  ( { m ,  X }  e.  ran  E  <-> 
m  e.  ( <. V ,  E >. Neighbors  X
) ) )
5554biimpa 484 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  m  e.  ( <. V ,  E >. Neighbors  X ) )
5655, 38sylibr 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  m  e.  D )
5752, 56jca 532 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ( ph  /\  m  e.  D ) )
58 preq1 3973 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  m  ->  { x ,  y }  =  { m ,  y } )
5958eleq1d 2509 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  m  ->  ( { x ,  y }  e.  ran  E  <->  { m ,  y }  e.  ran  E ) )
6059riotabidv 6073 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  m  ->  ( iota_ y  e.  N  {
x ,  y }  e.  ran  E )  =  ( iota_ y  e.  N  { m ,  y }  e.  ran  E ) )
6160cbvmptv 4402 . . . . . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . . . . . 21  |-  A  =  ( m  e.  D  |->  ( iota_ y  e.  N  { m ,  y }  e.  ran  E
) )
631, 2, 3, 4, 5, 6, 7, 62frgrancvvdeqlem6 30651 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  m  e.  D )  ->  { ( A `  m ) }  =  ( (
<. V ,  E >. Neighbors  m
)  i^i  N )
)
64 eleq2 2504 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  =  { ( A `  m ) }  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  <->  n  e.  { ( A `  m ) } ) )
6564eqcoms 2446 . . . . . . . . . . . . . . . . . . . . 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 3921 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  { ( A `
 m ) }  ->  n  =  ( A `  m ) )
6765, 66syl6bi 228 . . . . . . . . . . . . . . . . . . . 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 20 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  V USGrph  E )  /\  { m ,  X }  e.  ran  E )  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) )
6968expcom 435 . . . . . . . . . . . . . . . . . 18  |-  ( { m ,  X }  e.  ran  E  ->  (
( ph  /\  V USGrph  E
)  ->  ( n  e.  ( ( <. V ,  E >. Neighbors  m )  i^i  N
)  ->  n  =  ( A `  m ) ) ) )
7069ad2antll 728 . . . . . . . . . . . . . . . . 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 79 . . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . 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 432 . . . . . . . . . . . 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 532 . . . . . . . . . 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 434 . . . . . . . . 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 434 . . . . . . . 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 668 . . . . . . 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 429 . . . . . 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 1676 . . . . 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 2740 . . . 4  |-  ( E. m  e.  D  n  =  ( A `  m )  <->  E. m
( m  e.  D  /\  n  =  ( A `  m )
) )
8583, 84sylibr 212 . . 3  |-  ( (
ph  /\  n  e.  N )  ->  E. m  e.  D  n  =  ( A `  m ) )
8685ralrimiva 2818 . 2  |-  ( ph  ->  A. n  e.  N  E. m  e.  D  n  =  ( A `  m ) )
87 dffo3 5877 . 2  |-  ( A : D -onto-> N  <->  ( A : D --> N  /\  A. n  e.  N  E. m  e.  D  n  =  ( A `  m ) ) )
889, 86, 87sylanbrc 664 1  |-  ( ph  ->  A : D -onto-> N
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620    e/ wnel 2621   A.wral 2734   E.wrex 2735   E!wreu 2736    i^i cin 3346   {csn 3896   {cpr 3898   <.cop 3902   class class class wbr 4311    e. cmpt 4369   ran crn 4860   -->wf 5433   -onto->wfo 5435   ` cfv 5437   iota_crio 6070  (class class class)co 6110   USGrph cusg 23283   Neighbors cnbgra 23348   FriendGrph cfrgra 30603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-hash 12123  df-usgra 23285  df-nbgra 23351  df-frgra 30604
This theorem is referenced by:  frgrancvvdeqlem8  30656
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