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Theorem ssenen 6711
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 6607 . . 3  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1odm 5337 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
3 vex 2661 . . . . . . . 8  |-  f  e. 
_V
43dmex 4773 . . . . . . 7  |-  dom  f  e.  _V
52, 4syl6eqelr 2207 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
6 pwexg 4072 . . . . . 6  |-  ( A  e.  _V  ->  ~P A  e.  _V )
7 inex1g 4032 . . . . . 6  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  {
x  |  x  ~~  C } )  e.  _V )
85, 6, 73syl 17 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  e.  _V )
9 f1ofo 5340 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
10 forn 5316 . . . . . . . 8  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
119, 10syl 14 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
123rnex 4774 . . . . . . 7  |-  ran  f  e.  _V
1311, 12syl6eqelr 2207 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
14 pwexg 4072 . . . . . 6  |-  ( B  e.  _V  ->  ~P B  e.  _V )
15 inex1g 4032 . . . . . 6  |-  ( ~P B  e.  _V  ->  ( ~P B  i^i  {
x  |  x  ~~  C } )  e.  _V )
1613, 14, 153syl 17 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ( ~P B  i^i  { x  |  x  ~~  C }
)  e.  _V )
17 f1of1 5332 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  ->  f : A -1-1-> B )
1817adantr 272 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
f : A -1-1-> B
)
1913adantr 272 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  ->  B  e.  _V )
20 simpr 109 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  C_  A )
21 vex 2661 . . . . . . . . . . 11  |-  y  e. 
_V
2221a1i 9 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  e.  _V )
23 f1imaen2g 6653 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-> B  /\  B  e.  _V )  /\  ( y  C_  A  /\  y  e.  _V ) )  ->  (
f " y ) 
~~  y )
2418, 19, 20, 22, 23syl22anc 1200 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( f " y
)  ~~  y )
25 entr 6644 . . . . . . . . 9  |-  ( ( ( f " y
)  ~~  y  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
2624, 25sylan 279 . . . . . . . 8  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  y  ~~  C
)  ->  ( f " y )  ~~  C )
2726expl 373 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
)
28 imassrn 4860 . . . . . . . . 9  |-  ( f
" y )  C_  ran  f
2928, 10sseqtrid 3115 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( f " y
)  C_  B )
309, 29syl 14 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f " y )  C_  B )
3127, 30jctild 312 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
) )
32 elin 3227 . . . . . . 7  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  e.  ~P A  /\  y  e.  {
x  |  x  ~~  C } ) )
3321elpw 3484 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
34 breq1 3900 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  C  <->  y  ~~  C ) )
3521, 34elab 2800 . . . . . . . 8  |-  ( y  e.  { x  |  x  ~~  C }  <->  y 
~~  C )
3633, 35anbi12i 453 . . . . . . 7  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  <->  ( y  C_  A  /\  y  ~~  C
) )
3732, 36bitri 183 . . . . . 6  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  C_  A  /\  y  ~~  C ) )
38 elin 3227 . . . . . . 7  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  e.  ~P B  /\  ( f "
y )  e.  {
x  |  x  ~~  C } ) )
393imaex 4862 . . . . . . . . 9  |-  ( f
" y )  e. 
_V
4039elpw 3484 . . . . . . . 8  |-  ( ( f " y )  e.  ~P B  <->  ( f " y )  C_  B )
41 breq1 3900 . . . . . . . . 9  |-  ( x  =  ( f "
y )  ->  (
x  ~~  C  <->  ( f " y )  ~~  C ) )
4239, 41elab 2800 . . . . . . . 8  |-  ( ( f " y )  e.  { x  |  x  ~~  C }  <->  ( f " y ) 
~~  C )
4340, 42anbi12i 453 . . . . . . 7  |-  ( ( ( f " y
)  e.  ~P B  /\  ( f " y
)  e.  { x  |  x  ~~  C }
)  <->  ( ( f
" y )  C_  B  /\  ( f "
y )  ~~  C
) )
4438, 43bitri 183 . . . . . 6  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
)
4531, 37, 443imtr4g 204 . . . . 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 } ) ) )
46 f1ocnv 5346 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
47 f1of1 5332 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> A )
48 f1f1orn 5344 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-> A  ->  `' f : B -1-1-onto-> ran  `' f )
49 f1of1 5332 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> ran  `' f  ->  `' f : B -1-1-> ran  `' f )
5047, 48, 493syl 17 . . . . . . . . . . 11  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> ran  `' f )
51 vex 2661 . . . . . . . . . . . 12  |-  z  e. 
_V
5251f1imaen 6654 . . . . . . . . . . 11  |-  ( ( `' f : B -1-1-> ran  `' f  /\  z  C_  B )  ->  ( `' f " z
)  ~~  z )
5350, 52sylan 279 . . . . . . . . . 10  |-  ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  ->  ( `' f
" z )  ~~  z )
54 entr 6644 . . . . . . . . . 10  |-  ( ( ( `' f "
z )  ~~  z  /\  z  ~~  C )  ->  ( `' f
" z )  ~~  C )
5553, 54sylan 279 . . . . . . . . 9  |-  ( ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  /\  z  ~~  C )  ->  ( `' f " z
)  ~~  C )
5655expl 373 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( `' f " z )  ~~  C ) )
57 f1ofo 5340 . . . . . . . . 9  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -onto-> A )
58 imassrn 4860 . . . . . . . . . 10  |-  ( `' f " z ) 
C_  ran  `' f
59 forn 5316 . . . . . . . . . 10  |-  ( `' f : B -onto-> A  ->  ran  `' f  =  A )
6058, 59sseqtrid 3115 . . . . . . . . 9  |-  ( `' f : B -onto-> A  ->  ( `' f "
z )  C_  A
)
6157, 60syl 14 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( `' f "
z )  C_  A
)
6256, 61jctild 312 . . . . . . 7  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( ( `' f " z
)  C_  A  /\  ( `' f " z
)  ~~  C )
) )
6346, 62syl 14 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
z  C_  B  /\  z  ~~  C )  -> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) ) )
64 elin 3227 . . . . . . 7  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  e.  ~P B  /\  z  e.  {
x  |  x  ~~  C } ) )
6551elpw 3484 . . . . . . . 8  |-  ( z  e.  ~P B  <->  z  C_  B )
66 breq1 3900 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~~  C  <->  z  ~~  C ) )
6751, 66elab 2800 . . . . . . . 8  |-  ( z  e.  { x  |  x  ~~  C }  <->  z 
~~  C )
6865, 67anbi12i 453 . . . . . . 7  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  <->  ( z  C_  B  /\  z  ~~  C
) )
6964, 68bitri 183 . . . . . 6  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  C_  B  /\  z  ~~  C ) )
70 elin 3227 . . . . . . 7  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z )  e.  ~P A  /\  ( `' f " z
)  e.  { x  |  x  ~~  C }
) )
713cnvex 5045 . . . . . . . . . 10  |-  `' f  e.  _V
7271imaex 4862 . . . . . . . . 9  |-  ( `' f " z )  e.  _V
7372elpw 3484 . . . . . . . 8  |-  ( ( `' f " z
)  e.  ~P A  <->  ( `' f " z
)  C_  A )
74 breq1 3900 . . . . . . . . 9  |-  ( x  =  ( `' f
" z )  -> 
( x  ~~  C  <->  ( `' f " z
)  ~~  C )
)
7572, 74elab 2800 . . . . . . . 8  |-  ( ( `' f " z
)  e.  { x  |  x  ~~  C }  <->  ( `' f " z
)  ~~  C )
7673, 75anbi12i 453 . . . . . . 7  |-  ( ( ( `' f "
z )  e.  ~P A  /\  ( `' f
" z )  e. 
{ x  |  x 
~~  C } )  <-> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) )
7770, 76bitri 183 . . . . . 6  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z ) 
C_  A  /\  ( `' f " z
)  ~~  C )
)
7863, 69, 773imtr4g 204 . . . . 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 }
) ) )
79 simpl 108 . . . . . . . . . . 11  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  e.  ~P B )
8079elpwid 3489 . . . . . . . . . 10  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  C_  B )
8164, 80sylbi 120 . . . . . . . . 9  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  ->  z  C_  B
)
82 imaeq2 4845 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" z )  -> 
( f " y
)  =  ( f
" ( `' f
" z ) ) )
83 f1orel 5336 . . . . . . . . . . . . . . . 16  |-  ( f : A -1-1-onto-> B  ->  Rel  f )
84 dfrel2 4957 . . . . . . . . . . . . . . . 16  |-  ( Rel  f  <->  `' `' f  =  f
)
8583, 84sylib 121 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-onto-> B  ->  `' `' f  =  f )
8685imaeq1d 4848 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  ( `' `' f " ( `' f " z
) )  =  ( f " ( `' f " z ) ) )
8786adantr 272 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  ( f "
( `' f "
z ) ) )
8846, 47syl 14 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-> A )
89 f1imacnv 5350 . . . . . . . . . . . . . 14  |-  ( ( `' f : B -1-1-> A  /\  z  C_  B
)  ->  ( `' `' f " ( `' f " z
) )  =  z )
9088, 89sylan 279 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  z )
9187, 90eqtr3d 2150 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( f " ( `' f " z
) )  =  z )
9282, 91sylan9eqr 2170 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  (
f " y )  =  z )
9392eqcomd 2121 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  z  =  ( f "
y ) )
9493ex 114 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( y  =  ( `' f " z
)  ->  z  =  ( f " y
) ) )
9581, 94sylan2 282 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C }
) )  ->  (
y  =  ( `' f " z )  ->  z  =  ( f " y ) ) )
9695adantrl 467 . . . . . . 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 ) ) )
97 simpl 108 . . . . . . . . . . 11  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  e.  ~P A )
9897elpwid 3489 . . . . . . . . . 10  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  C_  A )
9932, 98sylbi 120 . . . . . . . . 9  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  ->  y  C_  A
)
100 imaeq2 4845 . . . . . . . . . . . 12  |-  ( z  =  ( f "
y )  ->  ( `' f " z
)  =  ( `' f " ( f
" y ) ) )
101 f1imacnv 5350 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  y  C_  A )  ->  ( `' f
" ( f "
y ) )  =  y )
10217, 101sylan 279 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( `' f "
( f " y
) )  =  y )
103100, 102sylan9eqr 2170 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  ( `' f " z )  =  y )
104103eqcomd 2121 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  y  =  ( `' f " z
) )
105104ex 114 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( z  =  ( f " y )  ->  y  =  ( `' f " z
) ) )
10699, 105sylan2 282 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  y  e.  ( ~P A  i^i  { x  |  x  ~~  C }
) )  ->  (
z  =  ( f
" y )  -> 
y  =  ( `' f " z ) ) )
107106adantrr 468 . . . . . . 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
) ) )
10896, 107impbid 128 . . . . . 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
) ) )
109108ex 114 . . . . 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 ) ) ) )
1108, 16, 45, 78, 109en3d 6629 . . . 4  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
111110exlimiv 1560 . . 3  |-  ( E. f  f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C } )  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
1121, 111sylbi 120 . 2  |-  ( A 
~~  B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
113 df-pw 3480 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
114113ineq1i 3241 . . 3  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)
115 inab 3312 . . 3  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
116114, 115eqtri 2136 . 2  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
117 df-pw 3480 . . . 4  |-  ~P B  =  { x  |  x 
C_  B }
118117ineq1i 3241 . . 3  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)
119 inab 3312 . . 3  |-  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
120118, 119eqtri 2136 . 2  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
121112, 116, 1203brtr3g 3929 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    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   _Vcvv 2658    i^i cin 3038    C_ wss 3039   ~Pcpw 3478   class class class wbr 3897   `'ccnv 4506   dom cdm 4507   ran crn 4508   "cima 4510   Rel wrel 4512   -1-1->wf1 5088   -onto->wfo 5089   -1-1-onto->wf1o 5090    ~~ cen 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-er 6395  df-en 6601
This theorem is referenced by: (None)
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