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Theorem ssenen 6557
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 6454 . . 3  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1odm 5251 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
3 vex 2622 . . . . . . . 8  |-  f  e. 
_V
43dmex 4695 . . . . . . 7  |-  dom  f  e.  _V
52, 4syl6eqelr 2179 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
6 pwexg 4013 . . . . . 6  |-  ( A  e.  _V  ->  ~P A  e.  _V )
7 inex1g 3973 . . . . . 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 5254 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
10 forn 5230 . . . . . . . 8  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
119, 10syl 14 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
123rnex 4696 . . . . . . 7  |-  ran  f  e.  _V
1311, 12syl6eqelr 2179 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
14 pwexg 4013 . . . . . 6  |-  ( B  e.  _V  ->  ~P B  e.  _V )
15 inex1g 3973 . . . . . 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 5246 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  ->  f : A -1-1-> B )
1817adantr 270 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
f : A -1-1-> B
)
1913adantr 270 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  ->  B  e.  _V )
20 simpr 108 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  C_  A )
21 vex 2622 . . . . . . . . . . 11  |-  y  e. 
_V
2221a1i 9 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  e.  _V )
23 f1imaen2g 6500 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-> B  /\  B  e.  _V )  /\  ( y  C_  A  /\  y  e.  _V ) )  ->  (
f " y ) 
~~  y )
2418, 19, 20, 22, 23syl22anc 1175 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( f " y
)  ~~  y )
25 entr 6491 . . . . . . . . 9  |-  ( ( ( f " y
)  ~~  y  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
2624, 25sylan 277 . . . . . . . 8  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  y  ~~  C
)  ->  ( f " y )  ~~  C )
2726expl 370 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
)
28 imassrn 4780 . . . . . . . . 9  |-  ( f
" y )  C_  ran  f
2928, 10syl5sseq 3074 . . . . . . . 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 309 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
) )
32 elin 3183 . . . . . . 7  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  e.  ~P A  /\  y  e.  {
x  |  x  ~~  C } ) )
3321elpw 3433 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
34 breq1 3846 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  C  <->  y  ~~  C ) )
3521, 34elab 2760 . . . . . . . 8  |-  ( y  e.  { x  |  x  ~~  C }  <->  y 
~~  C )
3633, 35anbi12i 448 . . . . . . 7  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  <->  ( y  C_  A  /\  y  ~~  C
) )
3732, 36bitri 182 . . . . . 6  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  C_  A  /\  y  ~~  C ) )
38 elin 3183 . . . . . . 7  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  e.  ~P B  /\  ( f "
y )  e.  {
x  |  x  ~~  C } ) )
393imaex 4782 . . . . . . . . 9  |-  ( f
" y )  e. 
_V
4039elpw 3433 . . . . . . . 8  |-  ( ( f " y )  e.  ~P B  <->  ( f " y )  C_  B )
41 breq1 3846 . . . . . . . . 9  |-  ( x  =  ( f "
y )  ->  (
x  ~~  C  <->  ( f " y )  ~~  C ) )
4239, 41elab 2760 . . . . . . . 8  |-  ( ( f " y )  e.  { x  |  x  ~~  C }  <->  ( f " y ) 
~~  C )
4340, 42anbi12i 448 . . . . . . 7  |-  ( ( ( f " y
)  e.  ~P B  /\  ( f " y
)  e.  { x  |  x  ~~  C }
)  <->  ( ( f
" y )  C_  B  /\  ( f "
y )  ~~  C
) )
4438, 43bitri 182 . . . . . 6  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
)
4531, 37, 443imtr4g 203 . . . . 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 5260 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
47 f1of1 5246 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> A )
48 f1f1orn 5258 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-> A  ->  `' f : B -1-1-onto-> ran  `' f )
49 f1of1 5246 . . . . . . . . . . . 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 2622 . . . . . . . . . . . 12  |-  z  e. 
_V
5251f1imaen 6501 . . . . . . . . . . 11  |-  ( ( `' f : B -1-1-> ran  `' f  /\  z  C_  B )  ->  ( `' f " z
)  ~~  z )
5350, 52sylan 277 . . . . . . . . . 10  |-  ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  ->  ( `' f
" z )  ~~  z )
54 entr 6491 . . . . . . . . . 10  |-  ( ( ( `' f "
z )  ~~  z  /\  z  ~~  C )  ->  ( `' f
" z )  ~~  C )
5553, 54sylan 277 . . . . . . . . 9  |-  ( ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  /\  z  ~~  C )  ->  ( `' f " z
)  ~~  C )
5655expl 370 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( `' f " z )  ~~  C ) )
57 f1ofo 5254 . . . . . . . . 9  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -onto-> A )
58 imassrn 4780 . . . . . . . . . 10  |-  ( `' f " z ) 
C_  ran  `' f
59 forn 5230 . . . . . . . . . 10  |-  ( `' f : B -onto-> A  ->  ran  `' f  =  A )
6058, 59syl5sseq 3074 . . . . . . . . 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 309 . . . . . . 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 3183 . . . . . . 7  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  e.  ~P B  /\  z  e.  {
x  |  x  ~~  C } ) )
6551elpw 3433 . . . . . . . 8  |-  ( z  e.  ~P B  <->  z  C_  B )
66 breq1 3846 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~~  C  <->  z  ~~  C ) )
6751, 66elab 2760 . . . . . . . 8  |-  ( z  e.  { x  |  x  ~~  C }  <->  z 
~~  C )
6865, 67anbi12i 448 . . . . . . 7  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  <->  ( z  C_  B  /\  z  ~~  C
) )
6964, 68bitri 182 . . . . . 6  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  C_  B  /\  z  ~~  C ) )
70 elin 3183 . . . . . . 7  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z )  e.  ~P A  /\  ( `' f " z
)  e.  { x  |  x  ~~  C }
) )
713cnvex 4964 . . . . . . . . . 10  |-  `' f  e.  _V
7271imaex 4782 . . . . . . . . 9  |-  ( `' f " z )  e.  _V
7372elpw 3433 . . . . . . . 8  |-  ( ( `' f " z
)  e.  ~P A  <->  ( `' f " z
)  C_  A )
74 breq1 3846 . . . . . . . . 9  |-  ( x  =  ( `' f
" z )  -> 
( x  ~~  C  <->  ( `' f " z
)  ~~  C )
)
7572, 74elab 2760 . . . . . . . 8  |-  ( ( `' f " z
)  e.  { x  |  x  ~~  C }  <->  ( `' f " z
)  ~~  C )
7673, 75anbi12i 448 . . . . . . 7  |-  ( ( ( `' f "
z )  e.  ~P A  /\  ( `' f
" z )  e. 
{ x  |  x 
~~  C } )  <-> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) )
7770, 76bitri 182 . . . . . 6  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z ) 
C_  A  /\  ( `' f " z
)  ~~  C )
)
7863, 69, 773imtr4g 203 . . . . 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 107 . . . . . . . . . . 11  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  e.  ~P B )
8079elpwid 3438 . . . . . . . . . 10  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  C_  B )
8164, 80sylbi 119 . . . . . . . . 9  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  ->  z  C_  B
)
82 imaeq2 4765 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" z )  -> 
( f " y
)  =  ( f
" ( `' f
" z ) ) )
83 f1orel 5250 . . . . . . . . . . . . . . . 16  |-  ( f : A -1-1-onto-> B  ->  Rel  f )
84 dfrel2 4876 . . . . . . . . . . . . . . . 16  |-  ( Rel  f  <->  `' `' f  =  f
)
8583, 84sylib 120 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-onto-> B  ->  `' `' f  =  f )
8685imaeq1d 4768 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  ( `' `' f " ( `' f " z
) )  =  ( f " ( `' f " z ) ) )
8786adantr 270 . . . . . . . . . . . . 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 5264 . . . . . . . . . . . . . 14  |-  ( ( `' f : B -1-1-> A  /\  z  C_  B
)  ->  ( `' `' f " ( `' f " z
) )  =  z )
9088, 89sylan 277 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  z )
9187, 90eqtr3d 2122 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( f " ( `' f " z
) )  =  z )
9282, 91sylan9eqr 2142 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  (
f " y )  =  z )
9392eqcomd 2093 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  z  =  ( f "
y ) )
9493ex 113 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( y  =  ( `' f " z
)  ->  z  =  ( f " y
) ) )
9581, 94sylan2 280 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C }
) )  ->  (
y  =  ( `' f " z )  ->  z  =  ( f " y ) ) )
9695adantrl 462 . . . . . . 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 107 . . . . . . . . . . 11  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  e.  ~P A )
9897elpwid 3438 . . . . . . . . . 10  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  C_  A )
9932, 98sylbi 119 . . . . . . . . 9  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  ->  y  C_  A
)
100 imaeq2 4765 . . . . . . . . . . . 12  |-  ( z  =  ( f "
y )  ->  ( `' f " z
)  =  ( `' f " ( f
" y ) ) )
101 f1imacnv 5264 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  y  C_  A )  ->  ( `' f
" ( f "
y ) )  =  y )
10217, 101sylan 277 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( `' f "
( f " y
) )  =  y )
103100, 102sylan9eqr 2142 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  ( `' f " z )  =  y )
104103eqcomd 2093 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  y  =  ( `' f " z
) )
105104ex 113 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( z  =  ( f " y )  ->  y  =  ( `' f " z
) ) )
10699, 105sylan2 280 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  y  e.  ( ~P A  i^i  { x  |  x  ~~  C }
) )  ->  (
z  =  ( f
" y )  -> 
y  =  ( `' f " z ) ) )
107106adantrr 463 . . . . . . 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 127 . . . . . 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 113 . . . . 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 6476 . . . 4  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
111110exlimiv 1534 . . 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 119 . 2  |-  ( A 
~~  B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
113 df-pw 3429 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
114113ineq1i 3197 . . 3  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)
115 inab 3267 . . 3  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
116114, 115eqtri 2108 . 2  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
117 df-pw 3429 . . . 4  |-  ~P B  =  { x  |  x 
C_  B }
118117ineq1i 3197 . . 3  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)
119 inab 3267 . . 3  |-  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
120118, 119eqtri 2108 . 2  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
121112, 116, 1203brtr3g 3874 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   _Vcvv 2619    i^i cin 2998    C_ wss 2999   ~Pcpw 3427   class class class wbr 3843   `'ccnv 4435   dom cdm 4436   ran crn 4437   "cima 4439   Rel wrel 4441   -1-1->wf1 5007   -onto->wfo 5008   -1-1-onto->wf1o 5009    ~~ cen 6445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-er 6282  df-en 6448
This theorem is referenced by: (None)
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