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Theorem bnj1408 28830
Description: Technical lemma for bnj1414 28831. 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
bnj1408.1  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
bnj1408.2  |-  C  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
bnj1408.3  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1408.4  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1408  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hints:    th( y)    ta( y)    B( y)    C( y)

Proof of Theorem bnj1408
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj1408.3 . . . 4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
21biimpri 197 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  th )
3 bnj1408.1 . . . . 5  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
43bnj1413 28829 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
5 simplll 734 . . . . . . . . 9  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  R  FrSe  A )
6 bnj213 28678 . . . . . . . . . . 11  |-  pred ( X ,  A ,  R )  C_  A
76sseli 3262 . . . . . . . . . 10  |-  ( z  e.  pred ( X ,  A ,  R )  ->  z  e.  A )
87adantl 452 . . . . . . . . 9  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  z  e.  A )
9 bnj906 28726 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
105, 8, 9syl2anc 642 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  pred ( z ,  A ,  R
)  C_  trCl ( z ,  A ,  R
) )
11 bnj1318 28819 . . . . . . . . . . 11  |-  ( y  =  z  ->  trCl (
y ,  A ,  R )  =  trCl ( z ,  A ,  R ) )
1211ssiun2s 4048 . . . . . . . . . 10  |-  ( z  e.  pred ( X ,  A ,  R )  ->  trCl ( z ,  A ,  R ) 
C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
13 ssun4 3429 . . . . . . . . . . 11  |-  (  trCl ( z ,  A ,  R )  C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  trCl (
z ,  A ,  R )  C_  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
1413, 3syl6sseqr 3311 . . . . . . . . . 10  |-  (  trCl ( z ,  A ,  R )  C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  trCl (
z ,  A ,  R )  C_  B
)
1512, 14syl 15 . . . . . . . . 9  |-  ( z  e.  pred ( X ,  A ,  R )  ->  trCl ( z ,  A ,  R ) 
C_  B )
1615adantl 452 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  trCl ( z ,  A ,  R
)  C_  B )
1710, 16sstrd 3275 . . . . . . 7  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  pred ( z ,  A ,  R
)  C_  B )
18 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  -> 
z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
1918bnj1405 28633 . . . . . . . . . 10  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  E. y  e.  pred  ( X ,  A ,  R ) z  e. 
trCl ( y ,  A ,  R ) )
20 biid 227 . . . . . . . . . 10  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  <->  ( (
( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) ) )
21 nfv 1624 . . . . . . . . . . . . 13  |-  F/ y ( R  FrSe  A  /\  X  e.  A
)
22 nfcv 2502 . . . . . . . . . . . . . . . 16  |-  F/_ y  pred ( X ,  A ,  R )
23 nfiu1 4035 . . . . . . . . . . . . . . . 16  |-  F/_ y U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R )
2422, 23nfun 3419 . . . . . . . . . . . . . . 15  |-  F/_ y
(  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
253, 24nfcxfr 2499 . . . . . . . . . . . . . 14  |-  F/_ y B
2625nfcri 2496 . . . . . . . . . . . . 13  |-  F/ y  z  e.  B
2721, 26nfan 1834 . . . . . . . . . . . 12  |-  F/ y ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )
2823nfcri 2496 . . . . . . . . . . . 12  |-  F/ y  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )
2927, 28nfan 1834 . . . . . . . . . . 11  |-  F/ y ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
3029nfri 1768 . . . . . . . . . 10  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  A. y ( ( ( R  FrSe  A  /\  X  e.  A )  /\  z  e.  B
)  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
3119, 20, 30bnj1521 28647 . . . . . . . . 9  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  E. y ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) ) )
32 simplll 734 . . . . . . . . . . . . 13  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  R  FrSe  A )
33323ad2ant1 977 . . . . . . . . . . . 12  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  R  FrSe  A )
34 bnj1147 28788 . . . . . . . . . . . . 13  |-  trCl (
y ,  A ,  R )  C_  A
35 simp3 958 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  -> 
z  e.  trCl (
y ,  A ,  R ) )
3634, 35bnj1213 28595 . . . . . . . . . . . 12  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  -> 
z  e.  A )
3733, 36, 9syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  pred ( z ,  A ,  R )  C_  trCl (
z ,  A ,  R ) )
38 simp2 957 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  -> 
y  e.  pred ( X ,  A ,  R ) )
396, 38bnj1213 28595 . . . . . . . . . . . 12  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  -> 
y  e.  A )
40 bnj1125 28786 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  y  e.  A  /\  z  e.  trCl ( y ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl (
y ,  A ,  R ) )
4133, 39, 35, 40syl3anc 1183 . . . . . . . . . . 11  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  trCl ( z ,  A ,  R )  C_  trCl (
y ,  A ,  R ) )
4237, 41sstrd 3275 . . . . . . . . . 10  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  pred ( z ,  A ,  R )  C_  trCl (
y ,  A ,  R ) )
43 ssiun2 4047 . . . . . . . . . . . 12  |-  ( y  e.  pred ( X ,  A ,  R )  ->  trCl ( y ,  A ,  R ) 
C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
44433ad2ant2 978 . . . . . . . . . . 11  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  trCl ( y ,  A ,  R )  C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
45 ssun4 3429 . . . . . . . . . . . 12  |-  (  trCl ( y ,  A ,  R )  C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
4645, 3syl6sseqr 3311 . . . . . . . . . . 11  |-  (  trCl ( y ,  A ,  R )  C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  B
)
4744, 46syl 15 . . . . . . . . . 10  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  trCl ( y ,  A ,  R )  C_  B
)
4842, 47sstrd 3275 . . . . . . . . 9  |-  ( ( ( ( ( R 
FrSe  A  /\  X  e.  A )  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  /\  y  e.  pred ( X ,  A ,  R
)  /\  z  e.  trCl ( y ,  A ,  R ) )  ->  pred ( z ,  A ,  R )  C_  B
)
4931, 48bnj593 28538 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  E. y  pred ( z ,  A ,  R
)  C_  B )
50 nfcv 2502 . . . . . . . . . 10  |-  F/_ y  pred ( z ,  A ,  R )
5150, 25nfss 3259 . . . . . . . . 9  |-  F/ y 
pred ( z ,  A ,  R ) 
C_  B
5251nfri 1768 . . . . . . . 8  |-  (  pred ( z ,  A ,  R )  C_  B  ->  A. y  pred (
z ,  A ,  R )  C_  B
)
5349, 52bnj1397 28631 . . . . . . 7  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  pred ( z ,  A ,  R )  C_  B
)
54 simpr 447 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  ->  z  e.  B )
553bnj1138 28584 . . . . . . . 8  |-  ( z  e.  B  <->  ( z  e.  pred ( X ,  A ,  R )  \/  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) ) )
5654, 55sylib 188 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  ->  (
z  e.  pred ( X ,  A ,  R )  \/  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
5717, 53, 56mpjaodan 761 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  ->  pred (
z ,  A ,  R )  C_  B
)
5857ralrimiva 2711 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. z  e.  B  pred ( z ,  A ,  R )  C_  B
)
59 df-bnj19 28486 . . . . 5  |-  (  TrFo ( B ,  A ,  R )  <->  A. z  e.  B  pred ( z ,  A ,  R
)  C_  B )
6058, 59sylibr 203 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
613bnj931 28566 . . . . 5  |-  pred ( X ,  A ,  R )  C_  B
6261a1i 10 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_  B )
63 bnj1408.4 . . . 4  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
644, 60, 62, 63syl3anbrc 1137 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
651, 63bnj1124 28782 . . 3  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
662, 64, 65syl2anc 642 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  C_  B )
67 bnj906 28726 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
68 iunss1 4018 . . . . 5  |-  (  pred ( X ,  A ,  R )  C_  trCl ( X ,  A ,  R )  ->  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
69 unss2 3434 . . . . 5  |-  ( U_ y  e.  pred  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) )  C_  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
7067, 68, 693syl 18 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  C_  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
71 bnj1408.2 . . . 4  |-  C  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
7270, 3, 713sstr4g 3305 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  C )
73 biid 227 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  <->  ( R  FrSe  A  /\  X  e.  A )
)
74 biid 227 . . . 4  |-  ( ( C  e.  _V  /\  TrFo ( C ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  C
)  <->  ( C  e. 
_V  /\  TrFo ( C ,  A ,  R
)  /\  pred ( X ,  A ,  R
)  C_  C )
)
7571, 73, 74bnj1136 28791 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  C )
7672, 75sseqtr4d 3301 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
7766, 76eqssd 3282 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873    u. cun 3236    C_ wss 3238   U_ciun 4007    predc-bnj14 28477    FrSe w-bnj15 28481    trClc-bnj18 28483    TrFow-bnj19 28485
This theorem is referenced by:  bnj1414  28831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-reg 7453  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6621  df-bnj17 28476  df-bnj14 28478  df-bnj13 28480  df-bnj15 28482  df-bnj18 28484  df-bnj19 28486
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