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Theorem ssenen 7764
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 7596 . . 3  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1odm 5832 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
3 vex 3034 . . . . . . . 8  |-  f  e. 
_V
43dmex 6745 . . . . . . 7  |-  dom  f  e.  _V
52, 4syl6eqelr 2558 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
6 pwexg 4585 . . . . . 6  |-  ( A  e.  _V  ->  ~P A  e.  _V )
7 inex1g 4539 . . . . . 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 5835 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
10 forn 5809 . . . . . . . 8  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
119, 10syl 17 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
123rnex 6746 . . . . . . 7  |-  ran  f  e.  _V
1311, 12syl6eqelr 2558 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
14 pwexg 4585 . . . . . 6  |-  ( B  e.  _V  ->  ~P B  e.  _V )
15 inex1g 4539 . . . . . 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 5827 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  ->  f : A -1-1-> B )
1817adantr 472 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
f : A -1-1-> B
)
1913adantr 472 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  ->  B  e.  _V )
20 simpr 468 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  C_  A )
21 vex 3034 . . . . . . . . . . 11  |-  y  e. 
_V
2221a1i 11 . . . . . . . . . 10  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
y  e.  _V )
23 f1imaen2g 7648 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-> B  /\  B  e.  _V )  /\  ( y  C_  A  /\  y  e.  _V ) )  ->  (
f " y ) 
~~  y )
2418, 19, 20, 22, 23syl22anc 1293 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( f " y
)  ~~  y )
25 entr 7639 . . . . . . . . 9  |-  ( ( ( f " y
)  ~~  y  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
2624, 25sylan 479 . . . . . . . 8  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  y  ~~  C
)  ->  ( f " y )  ~~  C )
2726expl 630 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( f " y
)  ~~  C )
)
28 imassrn 5185 . . . . . . . . 9  |-  ( f
" y )  C_  ran  f
2928, 10syl5sseq 3466 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( f " y
)  C_  B )
309, 29syl 17 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f " y )  C_  B )
3127, 30jctild 552 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
y  C_  A  /\  y  ~~  C )  -> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
) )
32 elin 3608 . . . . . . 7  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  e.  ~P A  /\  y  e.  {
x  |  x  ~~  C } ) )
3321elpw 3948 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
34 breq1 4398 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  C  <->  y  ~~  C ) )
3521, 34elab 3173 . . . . . . . 8  |-  ( y  e.  { x  |  x  ~~  C }  <->  y 
~~  C )
3633, 35anbi12i 711 . . . . . . 7  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  <->  ( y  C_  A  /\  y  ~~  C
) )
3732, 36bitri 257 . . . . . 6  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  <-> 
( y  C_  A  /\  y  ~~  C ) )
38 elin 3608 . . . . . . 7  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  e.  ~P B  /\  ( f "
y )  e.  {
x  |  x  ~~  C } ) )
39 imaexg 6749 . . . . . . . . . 10  |-  ( f  e.  _V  ->  (
f " y )  e.  _V )
403, 39ax-mp 5 . . . . . . . . 9  |-  ( f
" y )  e. 
_V
4140elpw 3948 . . . . . . . 8  |-  ( ( f " y )  e.  ~P B  <->  ( f " y )  C_  B )
42 breq1 4398 . . . . . . . . 9  |-  ( x  =  ( f "
y )  ->  (
x  ~~  C  <->  ( f " y )  ~~  C ) )
4340, 42elab 3173 . . . . . . . 8  |-  ( ( f " y )  e.  { x  |  x  ~~  C }  <->  ( f " y ) 
~~  C )
4441, 43anbi12i 711 . . . . . . 7  |-  ( ( ( f " y
)  e.  ~P B  /\  ( f " y
)  e.  { x  |  x  ~~  C }
)  <->  ( ( f
" y )  C_  B  /\  ( f "
y )  ~~  C
) )
4538, 44bitri 257 . . . . . 6  |-  ( ( f " y )  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( ( f "
y )  C_  B  /\  ( f " y
)  ~~  C )
)
4631, 37, 453imtr4g 278 . . . . 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 5840 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
48 f1of1 5827 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -1-1-> A )
49 f1f1orn 5839 . . . . . . . . . . . 12  |-  ( `' f : B -1-1-> A  ->  `' f : B -1-1-onto-> ran  `' f )
50 f1of1 5827 . . . . . . . . . . . 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 3034 . . . . . . . . . . . 12  |-  z  e. 
_V
5352f1imaen 7649 . . . . . . . . . . 11  |-  ( ( `' f : B -1-1-> ran  `' f  /\  z  C_  B )  ->  ( `' f " z
)  ~~  z )
5451, 53sylan 479 . . . . . . . . . 10  |-  ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  ->  ( `' f
" z )  ~~  z )
55 entr 7639 . . . . . . . . . 10  |-  ( ( ( `' f "
z )  ~~  z  /\  z  ~~  C )  ->  ( `' f
" z )  ~~  C )
5654, 55sylan 479 . . . . . . . . 9  |-  ( ( ( `' f : B -1-1-onto-> A  /\  z  C_  B )  /\  z  ~~  C )  ->  ( `' f " z
)  ~~  C )
5756expl 630 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( `' f " z )  ~~  C ) )
58 f1ofo 5835 . . . . . . . . 9  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B -onto-> A )
59 imassrn 5185 . . . . . . . . . 10  |-  ( `' f " z ) 
C_  ran  `' f
60 forn 5809 . . . . . . . . . 10  |-  ( `' f : B -onto-> A  ->  ran  `' f  =  A )
6159, 60syl5sseq 3466 . . . . . . . . 9  |-  ( `' f : B -onto-> A  ->  ( `' f "
z )  C_  A
)
6258, 61syl 17 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  -> 
( `' f "
z )  C_  A
)
6357, 62jctild 552 . . . . . . 7  |-  ( `' f : B -1-1-onto-> A  -> 
( ( z  C_  B  /\  z  ~~  C
)  ->  ( ( `' f " z
)  C_  A  /\  ( `' f " z
)  ~~  C )
) )
6447, 63syl 17 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( (
z  C_  B  /\  z  ~~  C )  -> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) ) )
65 elin 3608 . . . . . . 7  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  e.  ~P B  /\  z  e.  {
x  |  x  ~~  C } ) )
6652elpw 3948 . . . . . . . 8  |-  ( z  e.  ~P B  <->  z  C_  B )
67 breq1 4398 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  ~~  C  <->  z  ~~  C ) )
6852, 67elab 3173 . . . . . . . 8  |-  ( z  e.  { x  |  x  ~~  C }  <->  z 
~~  C )
6966, 68anbi12i 711 . . . . . . 7  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  <->  ( z  C_  B  /\  z  ~~  C
) )
7065, 69bitri 257 . . . . . 6  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  <-> 
( z  C_  B  /\  z  ~~  C ) )
71 elin 3608 . . . . . . 7  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z )  e.  ~P A  /\  ( `' f " z
)  e.  { x  |  x  ~~  C }
) )
723cnvex 6759 . . . . . . . . . 10  |-  `' f  e.  _V
73 imaexg 6749 . . . . . . . . . 10  |-  ( `' f  e.  _V  ->  ( `' f " z
)  e.  _V )
7472, 73ax-mp 5 . . . . . . . . 9  |-  ( `' f " z )  e.  _V
7574elpw 3948 . . . . . . . 8  |-  ( ( `' f " z
)  e.  ~P A  <->  ( `' f " z
)  C_  A )
76 breq1 4398 . . . . . . . . 9  |-  ( x  =  ( `' f
" z )  -> 
( x  ~~  C  <->  ( `' f " z
)  ~~  C )
)
7774, 76elab 3173 . . . . . . . 8  |-  ( ( `' f " z
)  e.  { x  |  x  ~~  C }  <->  ( `' f " z
)  ~~  C )
7875, 77anbi12i 711 . . . . . . 7  |-  ( ( ( `' f "
z )  e.  ~P A  /\  ( `' f
" z )  e. 
{ x  |  x 
~~  C } )  <-> 
( ( `' f
" z )  C_  A  /\  ( `' f
" z )  ~~  C ) )
7971, 78bitri 257 . . . . . 6  |-  ( ( `' f " z
)  e.  ( ~P A  i^i  { x  |  x  ~~  C }
)  <->  ( ( `' f " z ) 
C_  A  /\  ( `' f " z
)  ~~  C )
)
8064, 70, 793imtr4g 278 . . . . 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 464 . . . . . . . . . . 11  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  e.  ~P B )
8281elpwid 3952 . . . . . . . . . 10  |-  ( ( z  e.  ~P B  /\  z  e.  { x  |  x  ~~  C }
)  ->  z  C_  B )
8365, 82sylbi 200 . . . . . . . . 9  |-  ( z  e.  ( ~P B  i^i  { x  |  x 
~~  C } )  ->  z  C_  B
)
84 imaeq2 5170 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" z )  -> 
( f " y
)  =  ( f
" ( `' f
" z ) ) )
85 f1orel 5831 . . . . . . . . . . . . . . . 16  |-  ( f : A -1-1-onto-> B  ->  Rel  f )
86 dfrel2 5292 . . . . . . . . . . . . . . . 16  |-  ( Rel  f  <->  `' `' f  =  f
)
8785, 86sylib 201 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-onto-> B  ->  `' `' f  =  f )
8887imaeq1d 5173 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  ( `' `' f " ( `' f " z
) )  =  ( f " ( `' f " z ) ) )
8988adantr 472 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  ( f "
( `' f "
z ) ) )
9047, 48syl 17 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-> A )
91 f1imacnv 5844 . . . . . . . . . . . . . 14  |-  ( ( `' f : B -1-1-> A  /\  z  C_  B
)  ->  ( `' `' f " ( `' f " z
) )  =  z )
9290, 91sylan 479 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( `' `' f
" ( `' f
" z ) )  =  z )
9389, 92eqtr3d 2507 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( f " ( `' f " z
) )  =  z )
9484, 93sylan9eqr 2527 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  (
f " y )  =  z )
9594eqcomd 2477 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  z  C_  B )  /\  y  =  ( `' f " z
) )  ->  z  =  ( f "
y ) )
9695ex 441 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  z  C_  B )  -> 
( y  =  ( `' f " z
)  ->  z  =  ( f " y
) ) )
9783, 96sylan2 482 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  z  e.  ( ~P B  i^i  { x  |  x  ~~  C }
) )  ->  (
y  =  ( `' f " z )  ->  z  =  ( f " y ) ) )
9897adantrl 730 . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  e.  ~P A )
10099elpwid 3952 . . . . . . . . . 10  |-  ( ( y  e.  ~P A  /\  y  e.  { x  |  x  ~~  C }
)  ->  y  C_  A )
10132, 100sylbi 200 . . . . . . . . 9  |-  ( y  e.  ( ~P A  i^i  { x  |  x 
~~  C } )  ->  y  C_  A
)
102 imaeq2 5170 . . . . . . . . . . . 12  |-  ( z  =  ( f "
y )  ->  ( `' f " z
)  =  ( `' f " ( f
" y ) ) )
103 f1imacnv 5844 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  y  C_  A )  ->  ( `' f
" ( f "
y ) )  =  y )
10417, 103sylan 479 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( `' f "
( f " y
) )  =  y )
105102, 104sylan9eqr 2527 . . . . . . . . . . 11  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  ( `' f " z )  =  y )
106105eqcomd 2477 . . . . . . . . . 10  |-  ( ( ( f : A -1-1-onto-> B  /\  y  C_  A )  /\  z  =  ( f " y ) )  ->  y  =  ( `' f " z
) )
107106ex 441 . . . . . . . . 9  |-  ( ( f : A -1-1-onto-> B  /\  y  C_  A )  -> 
( z  =  ( f " y )  ->  y  =  ( `' f " z
) ) )
108101, 107sylan2 482 . . . . . . . 8  |-  ( ( f : A -1-1-onto-> B  /\  y  e.  ( ~P A  i^i  { x  |  x  ~~  C }
) )  ->  (
z  =  ( f
" y )  -> 
y  =  ( `' f " z ) ) )
109108adantrr 731 . . . . . . 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 195 . . . . . 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 441 . . . . 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 7624 . . . 4  |-  ( f : A -1-1-onto-> B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
113112exlimiv 1784 . . 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 200 . 2  |-  ( A 
~~  B  ->  ( ~P A  i^i  { x  |  x  ~~  C }
)  ~~  ( ~P B  i^i  { x  |  x  ~~  C }
) )
115 df-pw 3944 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
116115ineq1i 3621 . . 3  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)
117 inab 3702 . . 3  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
118116, 117eqtri 2493 . 2  |-  ( ~P A  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  A  /\  x  ~~  C
) }
119 df-pw 3944 . . . 4  |-  ~P B  =  { x  |  x 
C_  B }
120119ineq1i 3621 . . 3  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)
121 inab 3702 . . 3  |-  ( { x  |  x  C_  B }  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
122120, 121eqtri 2493 . 2  |-  ( ~P B  i^i  { x  |  x  ~~  C }
)  =  { x  |  ( x  C_  B  /\  x  ~~  C
) }
123114, 118, 1223brtr3g 4427 1  |-  ( A 
~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C
) }  ~~  {
x  |  ( x 
C_  B  /\  x  ~~  C ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   class class class wbr 4395   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Rel wrel 4844   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588    ~~ cen 7584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588
This theorem is referenced by:  infmap2  8666
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