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Theorem opsrtoslem2 16242
Description: Lemma for opsrtos 16243. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
opsrso.o  |-  O  =  ( ( I ordPwSer  R
) `  T )
opsrso.i  |-  ( ph  ->  I  e.  V )
opsrso.r  |-  ( ph  ->  R  e. Toset )
opsrso.t  |-  ( ph  ->  T  C_  ( I  X.  I ) )
opsrso.w  |-  ( ph  ->  T  We  I )
opsrtoslem.s  |-  S  =  ( I mPwSer  R )
opsrtoslem.b  |-  B  =  ( Base `  S
)
opsrtoslem.q  |-  .<  =  ( lt `  R )
opsrtoslem.c  |-  C  =  ( T  <bag  I )
opsrtoslem.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
opsrtoslem.ps  |-  ( ps  <->  E. z  e.  D  ( ( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) ) )
opsrtoslem.l  |-  .<_  =  ( le `  O )
Assertion
Ref Expression
opsrtoslem2  |-  ( ph  ->  O  e. Toset )
Distinct variable groups:    x, y, B    x, w, y, z, C    w, h, x, y, z, I    ph, w, x, y, z    w, D, x, y, z    w,  .< , x, y, z    w, R, x, y, z    w, T, x, y, z
Allowed substitution hints:    ph( h)    ps( x, y, z, w, h)    B( z, w, h)    C( h)    D( h)    R( h)    S( x, y, z, w, h)    .< ( h)    T( h)    .<_ ( x, y, z, w, h)    O( x, y, z, w, h)    V( x, y, z, w, h)

Proof of Theorem opsrtoslem2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrtoslem.d . . . . . . . . 9  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
2 ovex 5899 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
32rabex 4181 . . . . . . . . 9  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  e.  _V
41, 3eqeltri 2366 . . . . . . . 8  |-  D  e. 
_V
54a1i 10 . . . . . . 7  |-  ( ph  ->  D  e.  _V )
6 opsrtoslem.c . . . . . . . 8  |-  C  =  ( T  <bag  I )
7 opsrso.i . . . . . . . 8  |-  ( ph  ->  I  e.  V )
8 opsrso.t . . . . . . . . 9  |-  ( ph  ->  T  C_  ( I  X.  I ) )
9 xpexg 4816 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  I  e.  V )  ->  ( I  X.  I
)  e.  _V )
107, 7, 9syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( I  X.  I
)  e.  _V )
11 ssexg 4176 . . . . . . . . 9  |-  ( ( T  C_  ( I  X.  I )  /\  (
I  X.  I )  e.  _V )  ->  T  e.  _V )
128, 10, 11syl2anc 642 . . . . . . . 8  |-  ( ph  ->  T  e.  _V )
13 opsrso.w . . . . . . . 8  |-  ( ph  ->  T  We  I )
146, 1, 7, 12, 13ltbwe 16230 . . . . . . 7  |-  ( ph  ->  C  We  D )
15 opsrso.r . . . . . . . . 9  |-  ( ph  ->  R  e. Toset )
16 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
17 eqid 2296 . . . . . . . . . . 11  |-  ( le
`  R )  =  ( le `  R
)
18 opsrtoslem.q . . . . . . . . . . 11  |-  .<  =  ( lt `  R )
1916, 17, 18tosso 14158 . . . . . . . . . 10  |-  ( R  e. Toset  ->  ( R  e. Toset  <->  ( 
.<  Or  ( Base `  R
)  /\  (  _I  |`  ( Base `  R
) )  C_  ( le `  R ) ) ) )
2019ibi 232 . . . . . . . . 9  |-  ( R  e. Toset  ->  (  .<  Or  ( Base `  R )  /\  (  _I  |`  ( Base `  R ) )  C_  ( le `  R ) ) )
2115, 20syl 15 . . . . . . . 8  |-  ( ph  ->  (  .<  Or  ( Base `  R )  /\  (  _I  |`  ( Base `  R ) )  C_  ( le `  R ) ) )
2221simpld 445 . . . . . . 7  |-  ( ph  ->  .<  Or  ( Base `  R ) )
23 opsrtoslem.ps . . . . . . . . 9  |-  ( ps  <->  E. z  e.  D  ( ( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) ) )
2423opabbii 4099 . . . . . . . 8  |-  { <. x ,  y >.  |  ps }  =  { <. x ,  y >.  |  E. z  e.  D  (
( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
2524wemapso 7282 . . . . . . 7  |-  ( ( D  e.  _V  /\  C  We  D  /\  .<  Or  ( Base `  R
) )  ->  { <. x ,  y >.  |  ps }  Or  ( ( Base `  R )  ^m  D ) )
265, 14, 22, 25syl3anc 1182 . . . . . 6  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  Or  ( ( Base `  R
)  ^m  D )
)
27 opsrtoslem.s . . . . . . . 8  |-  S  =  ( I mPwSer  R )
28 opsrtoslem.b . . . . . . . 8  |-  B  =  ( Base `  S
)
2927, 16, 1, 28, 7psrbas 16140 . . . . . . 7  |-  ( ph  ->  B  =  ( (
Base `  R )  ^m  D ) )
30 soeq2 4350 . . . . . . 7  |-  ( B  =  ( ( Base `  R )  ^m  D
)  ->  ( { <. x ,  y >.  |  ps }  Or  B  <->  {
<. x ,  y >.  |  ps }  Or  (
( Base `  R )  ^m  D ) ) )
3129, 30syl 15 . . . . . 6  |-  ( ph  ->  ( { <. x ,  y >.  |  ps }  Or  B  <->  { <. x ,  y >.  |  ps }  Or  ( ( Base `  R )  ^m  D ) ) )
3226, 31mpbird 223 . . . . 5  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  Or  B )
33 soinxp 4770 . . . . 5  |-  ( {
<. x ,  y >.  |  ps }  Or  B  <->  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  Or  B
)
3432, 33sylib 188 . . . 4  |-  ( ph  ->  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  Or  B )
35 opsrso.o . . . . . . . 8  |-  O  =  ( ( I ordPwSer  R
) `  T )
36 fvex 5555 . . . . . . . 8  |-  ( ( I ordPwSer  R ) `  T
)  e.  _V
3735, 36eqeltri 2366 . . . . . . 7  |-  O  e. 
_V
38 opsrtoslem.l . . . . . . . 8  |-  .<_  =  ( le `  O )
39 eqid 2296 . . . . . . . 8  |-  ( lt
`  O )  =  ( lt `  O
)
4038, 39pltfval 14109 . . . . . . 7  |-  ( O  e.  _V  ->  ( lt `  O )  =  (  .<_  \  _I  )
)
4137, 40ax-mp 8 . . . . . 6  |-  ( lt
`  O )  =  (  .<_  \  _I  )
42 difundir 3435 . . . . . . . 8  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) )  \  _I  )  =  ( ( ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  \  _I  )  u.  ( (  _I  |`  B )  \  _I  ) )
43 resss 4995 . . . . . . . . . 10  |-  (  _I  |`  B )  C_  _I
44 ssdif0 3526 . . . . . . . . . 10  |-  ( (  _I  |`  B )  C_  _I  <->  ( (  _I  |`  B )  \  _I  )  =  (/) )
4543, 44mpbi 199 . . . . . . . . 9  |-  ( (  _I  |`  B )  \  _I  )  =  (/)
4645uneq2i 3339 . . . . . . . 8  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  \  _I  )  u.  (
(  _I  |`  B ) 
\  _I  ) )  =  ( ( ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  \  _I  )  u.  (/) )
47 un0 3492 . . . . . . . 8  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  \  _I  )  u.  (/) )  =  ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  \  _I  )
4842, 46, 473eqtri 2320 . . . . . . 7  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) )  \  _I  )  =  ( ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) 
\  _I  )
4935, 7, 15, 8, 13, 27, 28, 18, 6, 1, 23, 38opsrtoslem1 16241 . . . . . . . 8  |-  ( ph  -> 
.<_  =  ( ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) ) )
5049difeq1d 3306 . . . . . . 7  |-  ( ph  ->  (  .<_  \  _I  )  =  ( ( ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  u.  (  _I  |`  B ) ) 
\  _I  ) )
51 inss2 3403 . . . . . . . . . . . 12  |-  ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
52 relxp 4810 . . . . . . . . . . . 12  |-  Rel  ( B  X.  B )
53 relss 4791 . . . . . . . . . . . 12  |-  ( ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  C_  ( B  X.  B )  -> 
( Rel  ( B  X.  B )  ->  Rel  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) ) ) )
5451, 52, 53mp2 17 . . . . . . . . . . 11  |-  Rel  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )
5554a1i 10 . . . . . . . . . 10  |-  ( ph  ->  Rel  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) )
56 df-br 4040 . . . . . . . . . . . . . 14  |-  ( a  _I  b  <->  <. a ,  b >.  e.  _I  )
57 vex 2804 . . . . . . . . . . . . . . 15  |-  b  e. 
_V
5857ideq 4852 . . . . . . . . . . . . . 14  |-  ( a  _I  b  <->  a  =  b )
5956, 58bitr3i 242 . . . . . . . . . . . . 13  |-  ( <.
a ,  b >.  e.  _I  <->  a  =  b )
60 brin 4086 . . . . . . . . . . . . . . . . . 18  |-  ( a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  <-> 
( a { <. x ,  y >.  |  ps } a  /\  a
( B  X.  B
) a ) )
6160simprbi 450 . . . . . . . . . . . . . . . . 17  |-  ( a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  ->  a ( B  X.  B ) a )
62 brxp 4736 . . . . . . . . . . . . . . . . . 18  |-  ( a ( B  X.  B
) a  <->  ( a  e.  B  /\  a  e.  B ) )
6362simprbi 450 . . . . . . . . . . . . . . . . 17  |-  ( a ( B  X.  B
) a  ->  a  e.  B )
6461, 63syl 15 . . . . . . . . . . . . . . . 16  |-  ( a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  ->  a  e.  B
)
65 sonr 4351 . . . . . . . . . . . . . . . . 17  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  Or  B  /\  a  e.  B )  ->  -.  a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a )
6665ex 423 . . . . . . . . . . . . . . . 16  |-  ( ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  Or  B  ->  ( a  e.  B  ->  -.  a ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a ) )
6734, 64, 66syl2im 34 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( a ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  ->  -.  a
( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a ) )
6867pm2.01d 161 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  a ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a )
69 breq2 4043 . . . . . . . . . . . . . . . 16  |-  ( a  =  b  ->  (
a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  <-> 
a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) b ) )
70 df-br 4040 . . . . . . . . . . . . . . . 16  |-  ( a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) b  <->  <. a ,  b >.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) )
7169, 70syl6bb 252 . . . . . . . . . . . . . . 15  |-  ( a  =  b  ->  (
a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  <->  <. a ,  b >.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) ) )
7271notbid 285 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( -.  a ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) a  <->  -.  <. a ,  b
>.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) ) )
7368, 72syl5ibcom 211 . . . . . . . . . . . . 13  |-  ( ph  ->  ( a  =  b  ->  -.  <. a ,  b >.  e.  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) ) ) )
7459, 73syl5bi 208 . . . . . . . . . . . 12  |-  ( ph  ->  ( <. a ,  b
>.  e.  _I  ->  -.  <.
a ,  b >.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) ) )
7574con2d 107 . . . . . . . . . . 11  |-  ( ph  ->  ( <. a ,  b
>.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  ->  -.  <. a ,  b
>.  e.  _I  ) )
76 opex 4253 . . . . . . . . . . . 12  |-  <. a ,  b >.  e.  _V
77 eldif 3175 . . . . . . . . . . . 12  |-  ( <.
a ,  b >.  e.  ( _V  \  _I  ) 
<->  ( <. a ,  b
>.  e.  _V  /\  -.  <.
a ,  b >.  e.  _I  ) )
7876, 77mpbiran 884 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  e.  ( _V  \  _I  ) 
<->  -.  <. a ,  b
>.  e.  _I  )
7975, 78syl6ibr 218 . . . . . . . . . 10  |-  ( ph  ->  ( <. a ,  b
>.  e.  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  ->  <. a ,  b >.  e.  ( _V  \  _I  ) ) )
8055, 79relssdv 4795 . . . . . . . . 9  |-  ( ph  ->  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  C_  ( _V  \  _I  )
)
81 disj2 3515 . . . . . . . . 9  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  i^i 
_I  )  =  (/)  <->  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  C_  ( _V  \  _I  ) )
8280, 81sylibr 203 . . . . . . . 8  |-  ( ph  ->  ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  i^i 
_I  )  =  (/) )
83 disj3 3512 . . . . . . . 8  |-  ( ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  i^i 
_I  )  =  (/)  <->  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  =  ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  \  _I  ) )
8482, 83sylib 188 . . . . . . 7  |-  ( ph  ->  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  =  ( ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  \  _I  ) )
8548, 50, 843eqtr4a 2354 . . . . . 6  |-  ( ph  ->  (  .<_  \  _I  )  =  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) )
8641, 85syl5eq 2340 . . . . 5  |-  ( ph  ->  ( lt `  O
)  =  ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) ) )
87 soeq1 4349 . . . . 5  |-  ( ( lt `  O )  =  ( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  -> 
( ( lt `  O )  Or  B  <->  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  Or  B
) )
8886, 87syl 15 . . . 4  |-  ( ph  ->  ( ( lt `  O )  Or  B  <->  ( { <. x ,  y
>.  |  ps }  i^i  ( B  X.  B
) )  Or  B
) )
8934, 88mpbird 223 . . 3  |-  ( ph  ->  ( lt `  O
)  Or  B )
9027, 35, 8opsrbas 16236 . . . . 5  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  O ) )
9128, 90syl5eq 2340 . . . 4  |-  ( ph  ->  B  =  ( Base `  O ) )
92 soeq2 4350 . . . 4  |-  ( B  =  ( Base `  O
)  ->  ( ( lt `  O )  Or  B  <->  ( lt `  O )  Or  ( Base `  O ) ) )
9391, 92syl 15 . . 3  |-  ( ph  ->  ( ( lt `  O )  Or  B  <->  ( lt `  O )  Or  ( Base `  O
) ) )
9489, 93mpbid 201 . 2  |-  ( ph  ->  ( lt `  O
)  Or  ( Base `  O ) )
9591reseq2d 4971 . . . 4  |-  ( ph  ->  (  _I  |`  B )  =  (  _I  |`  ( Base `  O ) ) )
96 ssun2 3352 . . . . 5  |-  (  _I  |`  B )  C_  (
( { <. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) )
9796a1i 10 . . . 4  |-  ( ph  ->  (  _I  |`  B ) 
C_  ( ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) ) )
9895, 97eqsstr3d 3226 . . 3  |-  ( ph  ->  (  _I  |`  ( Base `  O ) ) 
C_  ( ( {
<. x ,  y >.  |  ps }  i^i  ( B  X.  B ) )  u.  (  _I  |`  B ) ) )
9998, 49sseqtr4d 3228 . 2  |-  ( ph  ->  (  _I  |`  ( Base `  O ) ) 
C_  .<_  )
100 eqid 2296 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
101100, 38, 39tosso 14158 . . 3  |-  ( O  e.  _V  ->  ( O  e. Toset  <->  ( ( lt
`  O )  Or  ( Base `  O
)  /\  (  _I  |`  ( Base `  O
) )  C_  .<_  ) ) )
10237, 101ax-mp 8 . 2  |-  ( O  e. Toset 
<->  ( ( lt `  O )  Or  ( Base `  O )  /\  (  _I  |`  ( Base `  O ) )  C_  .<_  ) )
10394, 99, 102sylanbrc 645 1  |-  ( ph  ->  O  e. Toset )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039   {copab 4092    _I cid 4320    Or wor 4329    We wwe 4367    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708   Rel wrel 4710   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   NNcn 9762   NN0cn0 9981   Basecbs 13164   lecple 13231   ltcplt 14091  Tosetctos 14155   mPwSer cmps 16103    <bag cltb 16110   ordPwSer copws 16111
This theorem is referenced by:  opsrtos  16243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-poset 14096  df-plt 14108  df-toset 14156  df-psr 16114  df-ltbag 16121  df-opsr 16122
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