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Theorem bnj1450 29859
Description: Technical lemma for bnj60 29871. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1450.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1450.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1450.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1450.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1450.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1450.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1450.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1450.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1450.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1450.10  |-  P  = 
U. H
bnj1450.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1450.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1450.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1450.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1450.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1450.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1450.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1450.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
bnj1450.19  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
bnj1450.20  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
bnj1450.21  |-  ( si  <->  ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
bnj1450.22  |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
bnj1450.23  |-  X  = 
<. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
Assertion
Ref Expression
bnj1450  |-  ( ze 
->  ( Q `  z
)  =  ( G `
 W ) )
Distinct variable groups:    A, d,
f, x, y, z    B, f    y, D    E, d, f, y    G, d, f, x, y, z    R, d, f, x, y, z    x, X    z, Y    ps, y
Allowed substitution hints:    ph( x, y, z, f, d)    ps( x, z, f, d)    ch( x, y, z, f, d)    th( x, y, z, f, d)    ta( x, y, z, f, d)    et( x, y, z, f, d)    ze( x, y, z, f, d)    si( x, y, z, f, d)    rh( x, y, z, f, d)    B( x, y, z, d)    C( x, y, z, f, d)    D( x, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    E( x, z)    H( x, y, z, f, d)    W( x, y, z, f, d)    X( y, z, f, d)    Y( x, y, f, d)    Z( x, y, z, f, d)    ta'( x, y, z, f, d)

Proof of Theorem bnj1450
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1450.19 . . . . . . . . 9  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
21simprbi 466 . . . . . . . 8  |-  ( ze 
->  z  e.  trCl ( x ,  A ,  R ) )
3 bnj1450.17 . . . . . . . . . 10  |-  ( th  <->  ( ch  /\  z  e.  E ) )
4 bnj1450.15 . . . . . . . . . . 11  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
5 fndm 5675 . . . . . . . . . . 11  |-  ( P  Fn  trCl ( x ,  A ,  R )  ->  dom  P  =  trCl ( x ,  A ,  R ) )
64, 5syl 17 . . . . . . . . . 10  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73, 6bnj832 29568 . . . . . . . . 9  |-  ( th 
->  dom  P  =  trCl ( x ,  A ,  R ) )
81, 7bnj832 29568 . . . . . . . 8  |-  ( ze 
->  dom  P  =  trCl ( x ,  A ,  R ) )
92, 8eleqtrrd 2532 . . . . . . 7  |-  ( ze 
->  z  e.  dom  P )
10 bnj1450.10 . . . . . . . 8  |-  P  = 
U. H
1110dmeqi 5036 . . . . . . 7  |-  dom  P  =  dom  U. H
129, 11syl6eleq 2539 . . . . . 6  |-  ( ze 
->  z  e.  dom  U. H )
13 bnj1450.9 . . . . . . . 8  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
1413bnj1317 29633 . . . . . . 7  |-  ( w  e.  H  ->  A. f  w  e.  H )
1514bnj1400 29647 . . . . . 6  |-  dom  U. H  =  U_ f  e.  H  dom  f
1612, 15syl6eleq 2539 . . . . 5  |-  ( ze 
->  z  e.  U_ f  e.  H  dom  f )
1716bnj1405 29648 . . . 4  |-  ( ze 
->  E. f  e.  H  z  e.  dom  f )
18 bnj1450.20 . . . 4  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
19 bnj1450.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
20 bnj1450.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
21 bnj1450.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
22 bnj1450.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
23 bnj1450.5 . . . . 5  |-  D  =  { x  e.  A  |  -.  E. f ta }
24 bnj1450.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
25 bnj1450.7 . . . . 5  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
26 bnj1450.8 . . . . 5  |-  ( ta'  <->  [. y  /  x ]. ta )
27 bnj1450.11 . . . . 5  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
28 bnj1450.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
29 bnj1450.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
30 bnj1450.14 . . . . 5  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
31 bnj1450.16 . . . . 5  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
32 bnj1450.18 . . . . 5  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
3319, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1bnj1449 29857 . . . 4  |-  ( ze 
->  A. f ze )
3417, 18, 33bnj1521 29662 . . 3  |-  ( ze 
->  E. f rh )
3513bnj1436 29651 . . . . . . . . . 10  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3618, 35bnj836 29571 . . . . . . . . 9  |-  ( rh 
->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3719, 20, 21, 22, 26bnj1373 29839 . . . . . . . . . 10  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3837rexbii 2889 . . . . . . . . 9  |-  ( E. y  e.  pred  (
x ,  A ,  R ) ta'  <->  E. y  e.  pred  ( x ,  A ,  R ) ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3936, 38sylib 200 . . . . . . . 8  |-  ( rh 
->  E. y  e.  pred  ( x ,  A ,  R ) ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
4039bnj1196 29606 . . . . . . 7  |-  ( rh 
->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) ) )
41 3anass 989 . . . . . . 7  |-  ( ( y  e.  pred (
x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  <-> 
( y  e.  pred ( x ,  A ,  R )  /\  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
4240, 41bnj1198 29607 . . . . . 6  |-  ( rh 
->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
43 bnj1450.21 . . . . . . 7  |-  ( si  <->  ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
44 bnj252 29508 . . . . . . 7  |-  ( ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )  <-> 
( rh  /\  (
y  e.  pred (
x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) ) )
4543, 44bitri 253 . . . . . 6  |-  ( si  <->  ( rh  /\  ( y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
4619, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18bnj1444 29852 . . . . . 6  |-  ( rh 
->  A. y rh )
4742, 45, 46bnj1340 29635 . . . . 5  |-  ( rh 
->  E. y si )
4821bnj1436 29651 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
4943, 48bnj771 29575 . . . . . . . . . 10  |-  ( si  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
5049bnj1196 29606 . . . . . . . . 9  |-  ( si  ->  E. d ( d  e.  B  /\  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
51 3anass 989 . . . . . . . . 9  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  ( d  e.  B  /\  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
5250, 51bnj1198 29607 . . . . . . . 8  |-  ( si  ->  E. d ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
53 bnj1450.22 . . . . . . . . 9  |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
54 bnj252 29508 . . . . . . . . 9  |-  ( ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  ( si  /\  ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) ) )
5553, 54bitri 253 . . . . . . . 8  |-  ( ph  <->  ( si  /\  ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
56 bnj1450.23 . . . . . . . . 9  |-  X  = 
<. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
5719, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18, 43, 53, 56bnj1445 29853 . . . . . . . 8  |-  ( si  ->  A. d si )
5852, 55, 57bnj1340 29635 . . . . . . 7  |-  ( si  ->  E. d ph )
5953bnj1254 29621 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
60 fveq2 5865 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
f `  x )  =  ( f `  z ) )
61 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  x  =  z )
62 bnj602 29726 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
6362reseq2d 5105 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( z ,  A ,  R ) ) )
6461, 63opeq12d 4174 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
6564, 20, 563eqtr4g 2510 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  Y  =  X )
6665fveq2d 5869 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( G `  Y )  =  ( G `  X ) )
6760, 66eqeq12d 2466 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( f `  z )  =  ( G `  X ) ) )
6867cbvralv 3019 . . . . . . . . . 10  |-  ( A. x  e.  d  (
f `  x )  =  ( G `  Y )  <->  A. z  e.  d  ( f `  z )  =  ( G `  X ) )
6959, 68sylib 200 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  d  ( f `  z
)  =  ( G `
 X ) )
7018simp3bi 1025 . . . . . . . . . . . 12  |-  ( rh 
->  z  e.  dom  f )
7143, 70bnj769 29573 . . . . . . . . . . 11  |-  ( si  ->  z  e.  dom  f
)
7253, 71bnj769 29573 . . . . . . . . . 10  |-  ( ph  ->  z  e.  dom  f
)
73 fndm 5675 . . . . . . . . . . 11  |-  ( f  Fn  d  ->  dom  f  =  d )
7453, 73bnj771 29575 . . . . . . . . . 10  |-  ( ph  ->  dom  f  =  d )
7572, 74eleqtrd 2531 . . . . . . . . 9  |-  ( ph  ->  z  e.  d )
7669, 75bnj1294 29629 . . . . . . . 8  |-  ( ph  ->  ( f `  z
)  =  ( G `
 X ) )
7731bnj930 29581 . . . . . . . . . . . . . 14  |-  ( ch 
->  Fun  Q )
783, 77bnj832 29568 . . . . . . . . . . . . 13  |-  ( th 
->  Fun  Q )
791, 78bnj832 29568 . . . . . . . . . . . 12  |-  ( ze 
->  Fun  Q )
8018, 79bnj835 29570 . . . . . . . . . . 11  |-  ( rh 
->  Fun  Q )
8143, 80bnj769 29573 . . . . . . . . . 10  |-  ( si  ->  Fun  Q )
8253, 81bnj769 29573 . . . . . . . . 9  |-  ( ph  ->  Fun  Q )
8318simp2bi 1024 . . . . . . . . . . . 12  |-  ( rh 
->  f  e.  H
)
8443, 83bnj769 29573 . . . . . . . . . . 11  |-  ( si  ->  f  e.  H )
8553, 84bnj769 29573 . . . . . . . . . 10  |-  ( ph  ->  f  e.  H )
86 elssuni 4227 . . . . . . . . . . 11  |-  ( f  e.  H  ->  f  C_ 
U. H )
8786, 10syl6sseqr 3479 . . . . . . . . . 10  |-  ( f  e.  H  ->  f  C_  P )
88 ssun3 3599 . . . . . . . . . . 11  |-  ( f 
C_  P  ->  f  C_  ( P  u.  { <. x ,  ( G `
 Z ) >. } ) )
8988, 28syl6sseqr 3479 . . . . . . . . . 10  |-  ( f 
C_  P  ->  f  C_  Q )
9085, 87, 893syl 18 . . . . . . . . 9  |-  ( ph  ->  f  C_  Q )
9182, 90, 72bnj1502 29659 . . . . . . . 8  |-  ( ph  ->  ( Q `  z
)  =  ( f `
 z ) )
9219bnj1517 29661 . . . . . . . . . . . . . . . 16  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
9353, 92bnj770 29574 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
9462sseq1d 3459 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
9594cbvralv 3019 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
9693, 95sylib 200 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d )
9796, 75bnj1294 29629 . . . . . . . . . . . . 13  |-  ( ph  ->  pred ( z ,  A ,  R ) 
C_  d )
9897, 74sseqtr4d 3469 . . . . . . . . . . . 12  |-  ( ph  ->  pred ( z ,  A ,  R ) 
C_  dom  f )
9982, 90, 98bnj1503 29660 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( f  |`  pred (
z ,  A ,  R ) ) )
10099opeq2d 4173 . . . . . . . . . 10  |-  ( ph  -> 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
)
101100, 29, 563eqtr4g 2510 . . . . . . . . 9  |-  ( ph  ->  W  =  X )
102101fveq2d 5869 . . . . . . . 8  |-  ( ph  ->  ( G `  W
)  =  ( G `
 X ) )
10376, 91, 1023eqtr4d 2495 . . . . . . 7  |-  ( ph  ->  ( Q `  z
)  =  ( G `
 W ) )
10458, 103bnj593 29555 . . . . . 6  |-  ( si  ->  E. d ( Q `
 z )  =  ( G `  W
) )
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 29854 . . . . . 6  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
106104, 105bnj1397 29646 . . . . 5  |-  ( si  ->  ( Q `  z
)  =  ( G `
 W ) )
10747, 106bnj593 29555 . . . 4  |-  ( rh 
->  E. y ( Q `
 z )  =  ( G `  W
) )
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 29855 . . . 4  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. y
( Q `  z
)  =  ( G `
 W ) )
109107, 108bnj1397 29646 . . 3  |-  ( rh 
->  ( Q `  z
)  =  ( G `
 W ) )
11034, 109bnj593 29555 . 2  |-  ( ze 
->  E. f ( Q `
 z )  =  ( G `  W
) )
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 29856 . 2  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( Q `  z
)  =  ( G `
 W ) )
112110, 111bnj1397 29646 1  |-  ( ze 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   [.wsbc 3267    u. cun 3402    C_ wss 3404   (/)c0 3731   {csn 3968   <.cop 3974   U.cuni 4198   U_ciun 4278   class class class wbr 4402   dom cdm 4834    |` cres 4836   Fun wfun 5576    Fn wfn 5577   ` cfv 5582    /\ w-bnj17 29491    predc-bnj14 29493    FrSe w-bnj15 29497    trClc-bnj18 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-bnj17 29492  df-bnj14 29494  df-bnj18 29500
This theorem is referenced by:  bnj1423  29860
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