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Theorem bnj1408 29066
Description: Technical lemma for bnj1414 29067. 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 29065 . . . 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 28914 . . . . . . . . . . 11  |-  pred ( X ,  A ,  R )  C_  A
76sseli 3176 . . . . . . . . . 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 28962 . . . . . . . . 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 29055 . . . . . . . . . . 11  |-  ( y  =  z  ->  trCl (
y ,  A ,  R )  =  trCl ( z ,  A ,  R ) )
1211ssiun2s 3946 . . . . . . . . . 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 3341 . . . . . . . . . . 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 3225 . . . . . . . . . 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 3189 . . . . . . 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 28869 . . . . . . . . . 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 1605 . . . . . . . . . . . . 13  |-  F/ y ( R  FrSe  A  /\  X  e.  A
)
22 nfcv 2419 . . . . . . . . . . . . . . . 16  |-  F/_ y  pred ( X ,  A ,  R )
23 nfiu1 3933 . . . . . . . . . . . . . . . 16  |-  F/_ y U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R )
2422, 23nfun 3331 . . . . . . . . . . . . . . 15  |-  F/_ y
(  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
253, 24nfcxfr 2416 . . . . . . . . . . . . . 14  |-  F/_ y B
2625nfcri 2413 . . . . . . . . . . . . 13  |-  F/ y  z  e.  B
2721, 26nfan 1771 . . . . . . . . . . . 12  |-  F/ y ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )
2823nfcri 2413 . . . . . . . . . . . 12  |-  F/ y  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )
2927, 28nfan 1771 . . . . . . . . . . 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 1742 . . . . . . . . . 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 28883 . . . . . . . . 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 976 . . . . . . . . . . . 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 29024 . . . . . . . . . . . . 13  |-  trCl (
y ,  A ,  R )  C_  A
35 simp3 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 ) )  -> 
z  e.  trCl (
y ,  A ,  R ) )
3634, 35bnj1213 28831 . . . . . . . . . . . 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 956 . . . . . . . . . . . . 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 28831 . . . . . . . . . . . 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 29022 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 3189 . . . . . . . . . 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 3945 . . . . . . . . . . . 12  |-  ( y  e.  pred ( X ,  A ,  R )  ->  trCl ( y ,  A ,  R ) 
C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
44433ad2ant2 977 . . . . . . . . . . 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 3341 . . . . . . . . . . . 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 3225 . . . . . . . . . . 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 3189 . . . . . . . . 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 28774 . . . . . . . 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 2419 . . . . . . . . . 10  |-  F/_ y  pred ( z ,  A ,  R )
5150, 25nfss 3173 . . . . . . . . 9  |-  F/ y 
pred ( z ,  A ,  R ) 
C_  B
5251nfri 1742 . . . . . . . 8  |-  (  pred ( z ,  A ,  R )  C_  B  ->  A. y  pred (
z ,  A ,  R )  C_  B
)
5349, 52bnj1397 28867 . . . . . . 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 28820 . . . . . . . 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 2626 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. z  e.  B  pred ( z ,  A ,  R )  C_  B
)
59 df-bnj19 28722 . . . . 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 28802 . . . . 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 1136 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
651, 63bnj1124 29018 . . 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 28962 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
68 iunss1 3916 . . . . 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 3346 . . . . 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 3219 . . 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 29027 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  C )
7672, 75sseqtr4d 3215 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
7766, 76eqssd 3196 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 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150    C_ wss 3152   U_ciun 3905    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1414  29067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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