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Theorem bnj1408 29111
Description: Technical lemma for bnj1414 29112. 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 198 . . 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 29110 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
5 simplll 735 . . . . . . . . 9  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  R  FrSe  A )
6 bnj213 28959 . . . . . . . . . . 11  |-  pred ( X ,  A ,  R )  C_  A
76sseli 3304 . . . . . . . . . 10  |-  ( z  e.  pred ( X ,  A ,  R )  ->  z  e.  A )
87adantl 453 . . . . . . . . 9  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  z  e.  A )
9 bnj906 29007 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
105, 8, 9syl2anc 643 . . . . . . . 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 29100 . . . . . . . . . . 11  |-  ( y  =  z  ->  trCl (
y ,  A ,  R )  =  trCl ( z ,  A ,  R ) )
1211ssiun2s 4095 . . . . . . . . . 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 3473 . . . . . . . . . . 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 3355 . . . . . . . . . 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 16 . . . . . . . . 9  |-  ( z  e.  pred ( X ,  A ,  R )  ->  trCl ( z ,  A ,  R ) 
C_  B )
1615adantl 453 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  trCl ( z ,  A ,  R
)  C_  B )
1710, 16sstrd 3318 . . . . . . 7  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  /\  z  e.  pred ( X ,  A ,  R )
)  ->  pred ( z ,  A ,  R
)  C_  B )
18 simpr 448 . . . . . . . . . . 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 28914 . . . . . . . . . 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 228 . . . . . . . . . 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 1626 . . . . . . . . . . . . 13  |-  F/ y ( R  FrSe  A  /\  X  e.  A
)
22 nfcv 2540 . . . . . . . . . . . . . . . 16  |-  F/_ y  pred ( X ,  A ,  R )
23 nfiu1 4081 . . . . . . . . . . . . . . . 16  |-  F/_ y U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R )
2422, 23nfun 3463 . . . . . . . . . . . . . . 15  |-  F/_ y
(  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
253, 24nfcxfr 2537 . . . . . . . . . . . . . 14  |-  F/_ y B
2625nfcri 2534 . . . . . . . . . . . . 13  |-  F/ y  z  e.  B
2721, 26nfan 1842 . . . . . . . . . . . 12  |-  F/ y ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )
2823nfcri 2534 . . . . . . . . . . . 12  |-  F/ y  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )
2927, 28nfan 1842 . . . . . . . . . . 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 1774 . . . . . . . . . 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 28928 . . . . . . . . 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 735 . . . . . . . . . . . . 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 978 . . . . . . . . . . . 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 29069 . . . . . . . . . . . . 13  |-  trCl (
y ,  A ,  R )  C_  A
35 simp3 959 . . . . . . . . . . . . 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 28876 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 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 ) )  -> 
y  e.  pred ( X ,  A ,  R ) )
396, 38bnj1213 28876 . . . . . . . . . . . 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 29067 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 3318 . . . . . . . . . 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 4094 . . . . . . . . . . . 12  |-  ( y  e.  pred ( X ,  A ,  R )  ->  trCl ( y ,  A ,  R ) 
C_  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
44433ad2ant2 979 . . . . . . . . . . 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 3473 . . . . . . . . . . . 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 3355 . . . . . . . . . . 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 16 . . . . . . . . . 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 3318 . . . . . . . . 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 28819 . . . . . . . 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 2540 . . . . . . . . . 10  |-  F/_ y  pred ( z ,  A ,  R )
5150, 25nfss 3301 . . . . . . . . 9  |-  F/ y 
pred ( z ,  A ,  R ) 
C_  B
5251nfri 1774 . . . . . . . 8  |-  (  pred ( z ,  A ,  R )  C_  B  ->  A. y  pred (
z ,  A ,  R )  C_  B
)
5349, 52bnj1397 28912 . . . . . . 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 448 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  ->  z  e.  B )
553bnj1138 28865 . . . . . . . 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 189 . . . . . . 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 762 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )  ->  pred (
z ,  A ,  R )  C_  B
)
5857ralrimiva 2749 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. z  e.  B  pred ( z ,  A ,  R )  C_  B
)
59 df-bnj19 28767 . . . . 5  |-  (  TrFo ( B ,  A ,  R )  <->  A. z  e.  B  pred ( z ,  A ,  R
)  C_  B )
6058, 59sylibr 204 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
613bnj931 28847 . . . . 5  |-  pred ( X ,  A ,  R )  C_  B
6261a1i 11 . . . 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 1138 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
651, 63bnj1124 29063 . . 3  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
662, 64, 65syl2anc 643 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  C_  B )
67 bnj906 29007 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
68 iunss1 4064 . . . . 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 3478 . . . . 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 19 . . . 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 3349 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  C )
73 biid 228 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  <->  ( R  FrSe  A  /\  X  e.  A )
)
74 biid 228 . . . 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 29072 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  C )
7672, 75sseqtr4d 3345 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
7766, 76eqssd 3325 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    u. cun 3278    C_ wss 3280   U_ciun 4053    predc-bnj14 28758    FrSe w-bnj15 28762    trClc-bnj18 28764    TrFow-bnj19 28766
This theorem is referenced by:  bnj1414  29112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-bnj17 28757  df-bnj14 28759  df-bnj13 28761  df-bnj15 28763  df-bnj18 28765  df-bnj19 28767
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