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Theorem bnj1408 29405
Description: Technical lemma for bnj1414 29406. 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 29404 . . . 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 29253 . . . . . . . . . . 11  |-  pred ( X ,  A ,  R )  C_  A
76sseli 3344 . . . . . . . . . 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 29301 . . . . . . . . 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 29394 . . . . . . . . . . 11  |-  ( y  =  z  ->  trCl (
y ,  A ,  R )  =  trCl ( z ,  A ,  R ) )
1211ssiun2s 4135 . . . . . . . . . 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 3513 . . . . . . . . . . 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 3395 . . . . . . . . . 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 3358 . . . . . . 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 29208 . . . . . . . . . 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 1629 . . . . . . . . . . . . 13  |-  F/ y ( R  FrSe  A  /\  X  e.  A
)
22 nfcv 2572 . . . . . . . . . . . . . . . 16  |-  F/_ y  pred ( X ,  A ,  R )
23 nfiu1 4121 . . . . . . . . . . . . . . . 16  |-  F/_ y U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R )
2422, 23nfun 3503 . . . . . . . . . . . . . . 15  |-  F/_ y
(  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
253, 24nfcxfr 2569 . . . . . . . . . . . . . 14  |-  F/_ y B
2625nfcri 2566 . . . . . . . . . . . . 13  |-  F/ y  z  e.  B
2721, 26nfan 1846 . . . . . . . . . . . 12  |-  F/ y ( ( R  FrSe  A  /\  X  e.  A
)  /\  z  e.  B )
2823nfcri 2566 . . . . . . . . . . . 12  |-  F/ y  z  e.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R )
2927, 28nfan 1846 . . . . . . . . . . 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 1778 . . . . . . . . . 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 29222 . . . . . . . . 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 29363 . . . . . . . . . . . . 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 29170 . . . . . . . . . . . 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 29170 . . . . . . . . . . . 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 29361 . . . . . . . . . . . 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 3358 . . . . . . . . . 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 4134 . . . . . . . . . . . 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 3513 . . . . . . . . . . . 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 3395 . . . . . . . . . . 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 3358 . . . . . . . . 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 29113 . . . . . . . 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 2572 . . . . . . . . . 10  |-  F/_ y  pred ( z ,  A ,  R )
5150, 25nfss 3341 . . . . . . . . 9  |-  F/ y 
pred ( z ,  A ,  R ) 
C_  B
5251nfri 1778 . . . . . . . 8  |-  (  pred ( z ,  A ,  R )  C_  B  ->  A. y  pred (
z ,  A ,  R )  C_  B
)
5349, 52bnj1397 29206 . . . . . . 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 29159 . . . . . . . 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 2789 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. z  e.  B  pred ( z ,  A ,  R )  C_  B
)
59 df-bnj19 29061 . . . . 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 29141 . . . . 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 29357 . . 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 29301 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
68 iunss1 4104 . . . . 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 3518 . . . . 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 3389 . . 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 29366 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  C )
7672, 75sseqtr4d 3385 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
7766, 76eqssd 3365 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 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    u. cun 3318    C_ wss 3320   U_ciun 4093    predc-bnj14 29052    FrSe w-bnj15 29056    trClc-bnj18 29058    TrFow-bnj19 29060
This theorem is referenced by:  bnj1414  29406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059  df-bnj19 29061
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