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Theorem ssenen 7035
Description: Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
ssenen  |-  ( A 
~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C
) }  ~~  {
x  |  ( x 
C_  B  /\  x  ~~  C ) } )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem ssenen
Dummy variables  y 
z  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 6871 . . 3  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1odm 5476 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
3 vex 2791 . . . . . . . 8  |-  f  e. 
_V
43dmex 4941 . . . . . . 7  |-  dom  f  e.  _V
52, 4syl6eqelr 2372 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
6 pwexg 4194 . . . . . 6  |-  ( A  e.  _V  ->  ~P A  e.  _V )
7 inex1g 4157 . . . . . 6  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  {
x  |  x  ~~  C } )  e.  _V )
85, 6, 73syl 18 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  e.  _V )
9 f1ofo 5479 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
10 forn 5454 . . . . . . . 8  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
119, 10syl 15 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
123rnex 4942 . . . . . . 7  |-  ran  f  e.  _V
1311, 12syl6eqelr 2372 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
14 pwexg 4194 . . . . . 6  |-  ( B  e.  _V  ->  ~P B  e.  _V )
15 inex1g 4157 . . . . . 6  |-  ( ~P B  e.  _V  ->  ( ~P B  i^i  {
x  |  x  ~~  C } )  e.  _V )
1613, 14, 153syl 18 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( ~P B  i^i  { x  |  x  ~~  C }
)  e.  _V )
17 f1of1 5471 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  ->  f : A -1-1-> B )
1817adantr 451 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
f : A -1-1-> B
)
1913adantr 451 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  ->  B  e.  _V )
20 simpr 447 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  C_  A )
21 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
2221a1i 10 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  e.  _V )
23 f1imaen2g 6922 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-> B  /\  B  e.  _V )  /\  ( y  C_  A  /\  y  e.  _V ) )  ->  (
f " y ) 
~~  y )
2418, 19, 20, 22, 23syl22anc 1183 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( f " y
)  ~~  y )
25 entr 6913 . . . . . . . . 9  |-  ( ( ( f " y
)  ~~  y  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
2624, 25sylan 457 . . . . . . . 8  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  y  ~~  C
)  ->  ( f " y )  ~~  C )
2726expl 601 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
)
28 imassrn 5025 . . . . . . . . 9  |-  ( f
" y )  C_  ran  f
2928, 10syl5sseq 3226 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( f " y
)  C_  B )
309, 29syl 15 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f " y )  C_  B )
3127, 30jctild 527 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
) )
32 elin 3358 . . . . . . 7  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  e.  ~P A  /\  y  e.  {
x  |  x  ~~  C } ) )
3321elpw 3631 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
34 breq1 4026 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  C  <->  y  ~~  C ) )
3521, 34elab 2914 . . . . . . . 8  |-  ( y  e.  { x  |  x  ~~  C }  <->  y 
~~  C )
3633, 35anbi12i 678 . . . . . . 7  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  <->  ( y  C_  A  /\  y  ~~  C
) )
3732, 36bitri 240 . . . . . 6  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  C_  A  /\  y  ~~  C ) )
38 elin 3358 . . . . . . 7  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  e.  ~P B  /\  ( f "
y )  e.  {
x  |  x  ~~  C } ) )
39 imaexg 5026 . . . . . . . . . 10  |-  ( f  e.  _V  ->  (
f " y )  e.  _V )
403, 39ax-mp 8 . . . . . . . . 9  |-  ( f
" y )  e. 
_V
4140elpw 3631 . . . . . . . 8  |-  ( ( f " y )  e.  ~P B  <->  ( f " y )  C_  B )
42 breq1 4026 . . . . . . . . 9  |-  ( x  =  ( f "
y )  ->  (
x  ~~  C  <->  ( f " y )  ~~  C ) )
4340, 42elab 2914 . . . . . . . 8  |-  ( ( f " y )  e.  { x  |  x  ~~  C }  <->  ( f " y ) 
~~  C )
4441, 43anbi12i 678 . . . . . . 7  |-  ( ( ( f " y
)  e.  ~P B  /\  ( f " y
)  e.  { x  |  x  ~~  C }
)  <->  ( ( f
" y )  C_  B  /\  ( f "
y )  ~~  C
) )
4538, 44bitri 240 . . . . . 6  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
)
4631, 37, 453imtr4g 261 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( y  e.  ( ~P A  i^i  { x  |  x  ~~  C } )  ->  (
f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } ) ) )
47 f1ocnv 5485 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
48 f1of1 5471 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> A )
49 f1f1orn 5483 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-> A  ->  `' f : B -1-1-onto-> ran  `' f )
50 f1of1 5471 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> ran  `' f  ->  `' f : B -1-1-> ran  `' f )
5148, 49, 503syl 18 . . . . . . . . . . 11  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> ran  `' f )
52 vex 2791 . . . . . . . . . . . 12  |-  z  e. 
_V
5352f1imaen 6923 . . . . . . . . . . 11  |-  ( ( `' f : B -1-1-> ran  `' f  /\  z  C_  B )  ->  ( `' f " z
)  ~~  z )
5451, 53sylan 457 . . . . . . . . . 10  |-  ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  ->  ( `' f
" z )  ~~  z )
55 entr 6913 . . . . . . . . . 10  |-  ( ( ( `' f "
z )  ~~  z  /\  z  ~~  C )  ->  ( `' f
" z )  ~~  C )
5654, 55sylan 457 . . . . . . . . 9  |-  ( ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  /\  z  ~~  C )  ->  ( `' f " z
)  ~~  C )
5756expl 601 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( `' f " z )  ~~  C ) )
58 f1ofo 5479 . . . . . . . . 9  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -onto-> A )
59 imassrn 5025 . . . . . . . . . 10  |-  ( `' f " z ) 
C_  ran  `' f
60 forn 5454 . . . . . . . . . 10  |-  ( `' f : B -onto-> A  ->  ran  `' f  =  A )
6159, 60syl5sseq 3226 . . . . . . . . 9  |-  ( `' f : B -onto-> A  ->  ( `' f "
z )  C_  A
)
6258, 61syl 15 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( `' f "
z )  C_  A
)
6357, 62jctild 527 . . . . . . 7  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( ( `' f " z
)  C_  A  /\  ( `' f " z
)  ~~  C )
) )
6447, 63syl 15 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
z  C_  B  /\  z  ~~  C )  -> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) ) )
65 elin 3358 . . . . . . 7  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  e.  ~P B  /\  z  e.  {
x  |  x  ~~  C } ) )
6652elpw 3631 . . . . . . . 8  |-  ( z  e.  ~P B  <->  z  C_  B )
67 breq1 4026 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~~  C  <->  z  ~~  C ) )
6852, 67elab 2914 . . . . . . . 8  |-  ( z  e.  { x  |  x  ~~  C }  <->  z 
~~  C )
6966, 68anbi12i 678 . . . . . . 7  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  <->  ( z  C_  B  /\  z  ~~  C
) )
7065, 69bitri 240 . . . . . 6  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  C_  B  /\  z  ~~  C ) )
71 elin 3358 . . . . . . 7  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z )  e.  ~P A  /\  ( `' f " z
)  e.  { x  |  x  ~~  C }
) )
723cnvex 5209 . . . . . . . . . 10  |-  `' f  e.  _V
73 imaexg 5026 . . . . . . . . . 10  |-  ( `' f  e.  _V  ->  ( `' f " z
)  e.  _V )
7472, 73ax-mp 8 . . . . . . . . 9  |-  ( `' f " z )  e.  _V
7574elpw 3631 . . . . . . . 8  |-  ( ( `' f " z
)  e.  ~P A  <->  ( `' f " z
)  C_  A )
76 breq1 4026 . . . . . . . . 9  |-  ( x  =  ( `' f
" z )  -> 
( x  ~~  C  <->  ( `' f " z
)  ~~  C )
)
7774, 76elab 2914 . . . . . . . 8  |-  ( ( `' f " z
)  e.  { x  |  x  ~~  C }  <->  ( `' f " z
)  ~~  C )
7875, 77anbi12i 678 . . . . . . 7  |-  ( ( ( `' f "
z )  e.  ~P A  /\  ( `' f
" z )  e. 
{ x  |  x 
~~  C } )  <-> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) )
7971, 78bitri 240 . . . . . 6  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z ) 
C_  A  /\  ( `' f " z
)  ~~  C )
)
8064, 70, 793imtr4g 261 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( z  e.  ( ~P B  i^i  { x  |  x  ~~  C } )  ->  ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
) ) )
81 simpl 443 . . . . . . . . . . 11  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  e.  ~P B )
8281, 66sylib 188 . . . . . . . . . 10  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  C_  B )
8365, 82sylbi 187 . . . . . . . . 9  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  ->  z  C_  B
)
84 imaeq2 5008 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" z )  -> 
( f " y
)  =  ( f
" ( `' f
" z ) ) )
85 f1orel 5475 . . . . . . . . . . . . . . . 16  |-  ( f : A -1-1-onto-> B  ->  Rel  f )
86 dfrel2 5124 . . . . . . . . . . . . . . . 16  |-  ( Rel  f  <->  `' `' f  =  f
)
8785, 86sylib 188 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-onto-> B  ->  `' `' f  =  f )
8887imaeq1d 5011 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  ( `' `' f " ( `' f " z
) )  =  ( f " ( `' f " z ) ) )
8988adantr 451 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  ( f "
( `' f "
z ) ) )
9047, 48syl 15 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-> A )
91 f1imacnv 5489 . . . . . . . . . . . . . 14  |-  ( ( `' f : B -1-1-> A  /\  z  C_  B
)  ->  ( `' `' f " ( `' f " z
) )  =  z )
9290, 91sylan 457 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  z )
9389, 92eqtr3d 2317 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( f " ( `' f " z
) )  =  z )
9484, 93sylan9eqr 2337 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  (
f " y )  =  z )
9594eqcomd 2288 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  z  =  ( f "
y ) )
9695ex 423 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( y  =  ( `' f " z
)  ->  z  =  ( f " y
) ) )
9783, 96sylan2 460 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C }
) )  ->  (
y  =  ( `' f " z )  ->  z  =  ( f " y ) ) )
9897adantrl 696 . . . . . . 7  |-  ( ( f : A -1-1-onto-> B  /\  ( y  e.  ( ~P A  i^i  {
x  |  x  ~~  C } )  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C } ) ) )  ->  ( y  =  ( `' f "
z )  ->  z  =  ( f "
y ) ) )
99 simpl 443 . . . . . . . . . . 11  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  e.  ~P A )
10099, 33sylib 188 . . . . . . . . . 10  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  C_  A )
10132, 100sylbi 187 . . . . . . . . 9  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  ->  y  C_  A
)
102 imaeq2 5008 . . . . . . . . . . . 12  |-  ( z  =  ( f "
y )  ->  ( `' f " z
)  =  ( `' f " ( f
" y ) ) )
103 f1imacnv 5489 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  y  C_  A )  ->  ( `' f
" ( f "
y ) )  =  y )
10417, 103sylan 457 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( `' f "
( f " y
) )  =  y )
105102, 104sylan9eqr 2337 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  ( `' f " z )  =  y )
106105eqcomd 2288 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  y  =  ( `' f " z
) )
107106ex 423 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( z  =  ( f " y )  ->  y  =  ( `' f " z
) ) )
108101, 107sylan2 460 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  y  e.  ( ~P A  i^i  { x  |  x  ~~  C }
) )  ->  (
z  =  ( f
" y )  -> 
y  =  ( `' f " z ) ) )
109108adantrr 697 . . . . . . 7  |-  ( ( f : A -1-1-onto-> B  /\  ( y  e.  ( ~P A  i^i  {
x  |  x  ~~  C } )  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C } ) ) )  ->  ( z  =  ( f " y
)  ->  y  =  ( `' f " z
) ) )
11098, 109impbid 183 . . . . . 6  |-  ( ( f : A -1-1-onto-> B  /\  ( y  e.  ( ~P A  i^i  {
x  |  x  ~~  C } )  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C } ) ) )  ->  ( y  =  ( `' f "
z )  <->  z  =  ( f " y
) ) )
111110ex 423 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( (
y  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  /\  z  e.  ( ~P B  i^i  {
x  |  x  ~~  C } ) )  -> 
( y  =  ( `' f " z
)  <->  z  =  ( f " y ) ) ) )
1128, 16, 46, 80, 111en3d 6898 . . . 4  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
113112exlimiv 1666 . . 3  |-  ( E. f  f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C } )  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
1141, 113sylbi 187 . 2  |-  ( A 
~~  B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
115 df-pw 3627 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
116115ineq1i 3366 . . 3  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)
117 inab 3436 . . 3  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
118116, 117eqtri 2303 . 2  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
119 df-pw 3627 . . . 4  |-  ~P B  =  { x  |  x 
C_  B }
120119ineq1i 3366 . . 3  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)
121 inab 3436 . . 3  |-  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
122120, 121eqtri 2303 . 2  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
123114, 118, 1223brtr3g 4054 1  |-  ( A 
~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C
) }  ~~  {
x  |  ( x 
C_  B  /\  x  ~~  C ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Rel wrel 4694   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  infmap2  7844
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-er 6660  df-en 6864
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