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Theorem bnj1450 32396
Description: Technical lemma for bnj60 32408. 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 464 . . . . . . . 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 5621 . . . . . . . . . . 11  |-  ( P  Fn  trCl ( x ,  A ,  R )  ->  dom  P  =  trCl ( x ,  A ,  R ) )
64, 5syl 16 . . . . . . . . . 10  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73, 6bnj832 32105 . . . . . . . . 9  |-  ( th 
->  dom  P  =  trCl ( x ,  A ,  R ) )
81, 7bnj832 32105 . . . . . . . 8  |-  ( ze 
->  dom  P  =  trCl ( x ,  A ,  R ) )
92, 8eleqtrrd 2545 . . . . . . 7  |-  ( ze 
->  z  e.  dom  P )
10 bnj1450.10 . . . . . . . 8  |-  P  = 
U. H
1110dmeqi 5152 . . . . . . 7  |-  dom  P  =  dom  U. H
129, 11syl6eleq 2552 . . . . . 6  |-  ( ze 
->  z  e.  dom  U. H )
13 bnj1450.9 . . . . . . . 8  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
1413bnj1317 32170 . . . . . . 7  |-  ( w  e.  H  ->  A. f  w  e.  H )
1514bnj1400 32184 . . . . . 6  |-  dom  U. H  =  U_ f  e.  H  dom  f
1612, 15syl6eleq 2552 . . . . 5  |-  ( ze 
->  z  e.  U_ f  e.  H  dom  f )
1716bnj1405 32185 . . . 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 32394 . . . 4  |-  ( ze 
->  A. f ze )
3417, 18, 33bnj1521 32199 . . 3  |-  ( ze 
->  E. f rh )
3513bnj1436 32188 . . . . . . . . . 10  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3618, 35bnj836 32108 . . . . . . . . 9  |-  ( rh 
->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3719, 20, 21, 22, 26bnj1373 32376 . . . . . . . . . 10  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3837rexbii 2862 . . . . . . . . 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 196 . . . . . . . 8  |-  ( rh 
->  E. y  e.  pred  ( x ,  A ,  R ) ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
4039bnj1196 32143 . . . . . . 7  |-  ( rh 
->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) ) )
41 3anass 969 . . . . . . 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 32144 . . . . . 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 32046 . . . . . . 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 249 . . . . . 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 32389 . . . . . 6  |-  ( rh 
->  A. y rh )
4742, 45, 46bnj1340 32172 . . . . 5  |-  ( rh 
->  E. y si )
4821bnj1436 32188 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
4943, 48bnj771 32112 . . . . . . . . . 10  |-  ( si  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
5049bnj1196 32143 . . . . . . . . 9  |-  ( si  ->  E. d ( d  e.  B  /\  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
51 3anass 969 . . . . . . . . 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 32144 . . . . . . . 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 32046 . . . . . . . . 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 249 . . . . . . . 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 32390 . . . . . . . 8  |-  ( si  ->  A. d si )
5852, 55, 57bnj1340 32172 . . . . . . 7  |-  ( si  ->  E. d ph )
5953bnj1254 32158 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
60 fveq2 5802 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
f `  x )  =  ( f `  z ) )
61 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  x  =  z )
62 bnj602 32263 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
6362reseq2d 5221 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( z ,  A ,  R ) ) )
6461, 63opeq12d 4178 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
6564, 20, 563eqtr4g 2520 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  Y  =  X )
6665fveq2d 5806 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( G `  Y )  =  ( G `  X ) )
6760, 66eqeq12d 2476 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( f `  z )  =  ( G `  X ) ) )
6867cbvralv 3053 . . . . . . . . . 10  |-  ( A. x  e.  d  (
f `  x )  =  ( G `  Y )  <->  A. z  e.  d  ( f `  z )  =  ( G `  X ) )
6959, 68sylib 196 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  d  ( f `  z
)  =  ( G `
 X ) )
7018simp3bi 1005 . . . . . . . . . . . 12  |-  ( rh 
->  z  e.  dom  f )
7143, 70bnj769 32110 . . . . . . . . . . 11  |-  ( si  ->  z  e.  dom  f
)
7253, 71bnj769 32110 . . . . . . . . . 10  |-  ( ph  ->  z  e.  dom  f
)
73 fndm 5621 . . . . . . . . . . 11  |-  ( f  Fn  d  ->  dom  f  =  d )
7453, 73bnj771 32112 . . . . . . . . . 10  |-  ( ph  ->  dom  f  =  d )
7572, 74eleqtrd 2544 . . . . . . . . 9  |-  ( ph  ->  z  e.  d )
7669, 75bnj1294 32166 . . . . . . . 8  |-  ( ph  ->  ( f `  z
)  =  ( G `
 X ) )
7731bnj930 32118 . . . . . . . . . . . . . 14  |-  ( ch 
->  Fun  Q )
783, 77bnj832 32105 . . . . . . . . . . . . 13  |-  ( th 
->  Fun  Q )
791, 78bnj832 32105 . . . . . . . . . . . 12  |-  ( ze 
->  Fun  Q )
8018, 79bnj835 32107 . . . . . . . . . . 11  |-  ( rh 
->  Fun  Q )
8143, 80bnj769 32110 . . . . . . . . . 10  |-  ( si  ->  Fun  Q )
8253, 81bnj769 32110 . . . . . . . . 9  |-  ( ph  ->  Fun  Q )
8318simp2bi 1004 . . . . . . . . . . . 12  |-  ( rh 
->  f  e.  H
)
8443, 83bnj769 32110 . . . . . . . . . . 11  |-  ( si  ->  f  e.  H )
8553, 84bnj769 32110 . . . . . . . . . 10  |-  ( ph  ->  f  e.  H )
86 elssuni 4232 . . . . . . . . . . 11  |-  ( f  e.  H  ->  f  C_ 
U. H )
8786, 10syl6sseqr 3514 . . . . . . . . . 10  |-  ( f  e.  H  ->  f  C_  P )
88 ssun3 3632 . . . . . . . . . . 11  |-  ( f 
C_  P  ->  f  C_  ( P  u.  { <. x ,  ( G `
 Z ) >. } ) )
8988, 28syl6sseqr 3514 . . . . . . . . . 10  |-  ( f 
C_  P  ->  f  C_  Q )
9085, 87, 893syl 20 . . . . . . . . 9  |-  ( ph  ->  f  C_  Q )
9182, 90, 72bnj1502 32196 . . . . . . . 8  |-  ( ph  ->  ( Q `  z
)  =  ( f `
 z ) )
9219bnj1517 32198 . . . . . . . . . . . . . . . 16  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
9353, 92bnj770 32111 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
9462sseq1d 3494 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
9594cbvralv 3053 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
9693, 95sylib 196 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d )
9796, 75bnj1294 32166 . . . . . . . . . . . . 13  |-  ( ph  ->  pred ( z ,  A ,  R ) 
C_  d )
9897, 74sseqtr4d 3504 . . . . . . . . . . . 12  |-  ( ph  ->  pred ( z ,  A ,  R ) 
C_  dom  f )
9982, 90, 98bnj1503 32197 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( f  |`  pred (
z ,  A ,  R ) ) )
10099opeq2d 4177 . . . . . . . . . 10  |-  ( ph  -> 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
)
101100, 29, 563eqtr4g 2520 . . . . . . . . 9  |-  ( ph  ->  W  =  X )
102101fveq2d 5806 . . . . . . . 8  |-  ( ph  ->  ( G `  W
)  =  ( G `
 X ) )
10376, 91, 1023eqtr4d 2505 . . . . . . 7  |-  ( ph  ->  ( Q `  z
)  =  ( G `
 W ) )
10458, 103bnj593 32092 . . . . . 6  |-  ( si  ->  E. d ( Q `
 z )  =  ( G `  W
) )
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 32391 . . . . . 6  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
106104, 105bnj1397 32183 . . . . 5  |-  ( si  ->  ( Q `  z
)  =  ( G `
 W ) )
10747, 106bnj593 32092 . . . 4  |-  ( rh 
->  E. y ( Q `
 z )  =  ( G `  W
) )
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 32392 . . . 4  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. y
( Q `  z
)  =  ( G `
 W ) )
109107, 108bnj1397 32183 . . 3  |-  ( rh 
->  ( Q `  z
)  =  ( G `
 W ) )
11034, 109bnj593 32092 . 2  |-  ( ze 
->  E. f ( Q `
 z )  =  ( G `  W
) )
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 32393 . 2  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( Q `  z
)  =  ( G `
 W ) )
112110, 111bnj1397 32183 1  |-  ( ze 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   [.wsbc 3294    u. cun 3437    C_ wss 3439   (/)c0 3748   {csn 3988   <.cop 3994   U.cuni 4202   U_ciun 4282   class class class wbr 4403   dom cdm 4951    |` cres 4953   Fun wfun 5523    Fn wfn 5524   ` cfv 5529    /\ w-bnj17 32029    predc-bnj14 32031    FrSe w-bnj15 32035    trClc-bnj18 32037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-bnj17 32030  df-bnj14 32032  df-bnj18 32038
This theorem is referenced by:  bnj1423  32397
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