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Theorem hashbc 11386
Description: The binomial coefficient counts the number of subsets of a finite set of a given size. (Contributed by Mario Carneiro, 13-Jul-2014.)
Assertion
Ref Expression
hashbc  |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A
)  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
Distinct variable groups:    x, A    x, K

Proof of Theorem hashbc
Dummy variables  j 
k  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5485 . . . . . 6  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
21oveq1d 5834 . . . . 5  |-  ( w  =  (/)  ->  ( (
# `  w )  _C  k )  =  ( ( # `  (/) )  _C  k ) )
3 pweq 3628 . . . . . . 7  |-  ( w  =  (/)  ->  ~P w  =  ~P (/) )
4 rabeq 2782 . . . . . . 7  |-  ( ~P w  =  ~P (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
53, 4syl 15 . . . . . 6  |-  ( w  =  (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
65fveq2d 5489 . . . . 5  |-  ( w  =  (/)  ->  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
72, 6eqeq12d 2297 . . . 4  |-  ( w  =  (/)  ->  ( ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) ) )
87ralbidv 2563 . . 3  |-  ( w  =  (/)  ->  ( A. k  e.  ZZ  (
( # `  w )  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) ) )
9 fveq2 5485 . . . . . 6  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
109oveq1d 5834 . . . . 5  |-  ( w  =  y  ->  (
( # `  w )  _C  k )  =  ( ( # `  y
)  _C  k ) )
11 pweq 3628 . . . . . . 7  |-  ( w  =  y  ->  ~P w  =  ~P y
)
12 rabeq 2782 . . . . . . 7  |-  ( ~P w  =  ~P y  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P y  |  (
# `  x )  =  k } )
1311, 12syl 15 . . . . . 6  |-  ( w  =  y  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  k } )
1413fveq2d 5489 . . . . 5  |-  ( w  =  y  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) )
1510, 14eqeq12d 2297 . . . 4  |-  ( w  =  y  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  k
)  =  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
1615ralbidv 2563 . . 3  |-  ( w  =  y  ->  ( A. k  e.  ZZ  ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
17 fveq2 5485 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  w
)  =  ( # `  ( y  u.  {
z } ) ) )
1817oveq1d 5834 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( # `  w )  _C  k
)  =  ( (
# `  ( y  u.  { z } ) )  _C  k ) )
19 pweq 3628 . . . . . . 7  |-  ( w  =  ( y  u. 
{ z } )  ->  ~P w  =  ~P ( y  u. 
{ z } ) )
20 rabeq 2782 . . . . . . 7  |-  ( ~P w  =  ~P (
y  u.  { z } )  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2119, 20syl 15 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  { x  e. 
~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2221fveq2d 5489 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } ) )
2318, 22eqeq12d 2297 . . . 4  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( (
# `  w )  _C  k )  =  (
# `  { x  e.  ~P w  |  (
# `  x )  =  k } )  <-> 
( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) ) )
2423ralbidv 2563 . . 3  |-  ( w  =  ( y  u. 
{ z } )  ->  ( A. k  e.  ZZ  ( ( # `  w )  _C  k
)  =  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) ) )
25 fveq2 5485 . . . . . 6  |-  ( w  =  A  ->  ( # `
 w )  =  ( # `  A
) )
2625oveq1d 5834 . . . . 5  |-  ( w  =  A  ->  (
( # `  w )  _C  k )  =  ( ( # `  A
)  _C  k ) )
27 pweq 3628 . . . . . . 7  |-  ( w  =  A  ->  ~P w  =  ~P A
)
28 rabeq 2782 . . . . . . 7  |-  ( ~P w  =  ~P A  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P A  |  (
# `  x )  =  k } )
2927, 28syl 15 . . . . . 6  |-  ( w  =  A  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  k } )
3029fveq2d 5489 . . . . 5  |-  ( w  =  A  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) )
3126, 30eqeq12d 2297 . . . 4  |-  ( w  =  A  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
3231ralbidv 2563 . . 3  |-  ( w  =  A  ->  ( A. k  e.  ZZ  ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  A. k  e.  ZZ  ( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
33 hash0 11350 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
3433a1i 10 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 (/) )  =  0 )
35 elfz1eq 10802 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
3634, 35oveq12d 5837 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  ( 0  _C  0 ) )
37 0nn0 9975 . . . . . . . . 9  |-  0  e.  NN0
38 bcn0 11318 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
4036, 39syl6eq 2331 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  1 )
41 pw0 3762 . . . . . . . . . 10  |-  ~P (/)  =  { (/)
}
4235eqcomd 2288 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 0 )  ->  0  =  k )
4341raleqi 2740 . . . . . . . . . . . . 13  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  A. x  e.  { (/) }  ( # `  x
)  =  k )
44 0ex 4150 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
45 fveq2 5485 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
4645, 33syl6eq 2331 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
4746eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( (
# `  x )  =  k  <->  0  =  k ) )
4844, 47ralsn 3674 . . . . . . . . . . . . 13  |-  ( A. x  e.  { (/) }  ( # `
 x )  =  k  <->  0  =  k )
4943, 48bitri 240 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  0  =  k )
5042, 49sylibr 203 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... 0 )  ->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
51 rabid2 2717 . . . . . . . . . . 11  |-  ( ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k }  <->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
5250, 51sylibr 203 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... 0 )  ->  ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k } )
5341, 52syl5reqr 2330 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  { x  e.  ~P (/)  |  ( # `
 x )  =  k }  =  { (/)
} )
5453fveq2d 5489 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  (
# `  { (/) } ) )
55 hashsng 11351 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( # `  { (/)
} )  =  1 )
5644, 55ax-mp 8 . . . . . . . 8  |-  ( # `  { (/) } )  =  1
5754, 56syl6eq 2331 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  1 )
5840, 57eqtr4d 2318 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
5958adantl 452 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  e.  ( 0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
6033oveq1i 5829 . . . . . 6  |-  ( (
# `  (/) )  _C  k )  =  ( 0  _C  k )
61 bcval3 11314 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  k  e.  ZZ  /\  -.  k  e.  ( 0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
6237, 61mp3an1 1264 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
63 id 19 . . . . . . . . . . . . . 14  |-  ( 0  =  k  ->  0  =  k )
64 0z 10030 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
65 elfz3 10801 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
6664, 65ax-mp 8 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 ... 0
)
6763, 66syl6eqelr 2372 . . . . . . . . . . . . 13  |-  ( 0  =  k  ->  k  e.  ( 0 ... 0
) )
6867con3i 127 . . . . . . . . . . . 12  |-  ( -.  k  e.  ( 0 ... 0 )  ->  -.  0  =  k
)
6968adantl 452 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  -.  0  =  k )
7041raleqi 2740 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  A. x  e.  { (/)
}  -.  ( # `  x )  =  k )
7147notbid 285 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( -.  ( # `  x
)  =  k  <->  -.  0  =  k ) )
7244, 71ralsn 3674 . . . . . . . . . . . 12  |-  ( A. x  e.  { (/) }  -.  ( # `  x )  =  k  <->  -.  0  =  k )
7370, 72bitri 240 . . . . . . . . . . 11  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  -.  0  =  k )
7469, 73sylibr 203 . . . . . . . . . 10  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  A. x  e.  ~P  (/) 
-.  ( # `  x
)  =  k )
75 rabeq0 3476 . . . . . . . . . 10  |-  ( { x  e.  ~P (/)  |  (
# `  x )  =  k }  =  (/)  <->  A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k )
7674, 75sylibr 203 . . . . . . . . 9  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  { x  e. 
~P (/)  |  ( # `  x )  =  k }  =  (/) )
7776fveq2d 5489 . . . . . . . 8  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  ( # `  (/) ) )
7877, 33syl6eq 2331 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  0 )
7962, 78eqtr4d 2318 . . . . . 6  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8060, 79syl5eq 2327 . . . . 5  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
8159, 80pm2.61dan 766 . . . 4  |-  ( k  e.  ZZ  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8281rgen 2608 . . 3  |-  A. k  e.  ZZ  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
83 oveq2 5827 . . . . . 6  |-  ( k  =  j  ->  (
( # `  y )  _C  k )  =  ( ( # `  y
)  _C  j ) )
84 eqeq2 2292 . . . . . . . . 9  |-  ( k  =  j  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  j ) )
8584rabbidv 2780 . . . . . . . 8  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  j } )
86 fveq2 5485 . . . . . . . . . 10  |-  ( x  =  z  ->  ( # `
 x )  =  ( # `  z
) )
8786eqeq1d 2291 . . . . . . . . 9  |-  ( x  =  z  ->  (
( # `  x )  =  j  <->  ( # `  z
)  =  j ) )
8887cbvrabv 2787 . . . . . . . 8  |-  { x  e.  ~P y  |  (
# `  x )  =  j }  =  { z  e.  ~P y  |  ( # `  z
)  =  j }
8985, 88syl6eq 2331 . . . . . . 7  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9089fveq2d 5489 . . . . . 6  |-  ( k  =  j  ->  ( # `
 { x  e. 
~P y  |  (
# `  x )  =  k } )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9183, 90eqeq12d 2297 . . . . 5  |-  ( k  =  j  ->  (
( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )
9291cbvralv 2764 . . . 4  |-  ( A. k  e.  ZZ  (
( # `  y )  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
93 simpll 730 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
y  e.  Fin )
94 simplr 731 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  -.  z  e.  y
)
95 simprr 733 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9688fveq2i 5488 . . . . . . . . . 10  |-  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  j } )  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9796eqeq2i 2293 . . . . . . . . 9  |-  ( ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9897ralbii 2567 . . . . . . . 8  |-  ( A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9995, 98sylibr 203 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  ->  A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } ) )
100 simprl 732 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
k  e.  ZZ )
10193, 94, 99, 100hashbclem 11385 . . . . . 6  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( k  e.  ZZ  /\  A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )  -> 
( ( # `  (
y  u.  { z } ) )  _C  k )  =  (
# `  { x  e.  ~P ( y  u. 
{ z } )  |  ( # `  x
)  =  k } ) )
102101expr 598 . . . . 5  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  k  e.  ZZ )  ->  ( A. j  e.  ZZ  ( ( # `  y
)  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } )  ->  ( ( # `
 ( y  u. 
{ z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
103102ralrimdva 2633 . . . 4  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( A. j  e.  ZZ  (
( # `  y )  _C  j )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } )  ->  A. k  e.  ZZ  ( ( # `  ( y  u.  {
z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
10492, 103syl5bi 208 . . 3  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( A. k  e.  ZZ  (
( # `  y )  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  ->  A. k  e.  ZZ  ( ( # `  ( y  u.  {
z } ) )  _C  k )  =  ( # `  {
x  e.  ~P (
y  u.  { z } )  |  (
# `  x )  =  k } ) ) )
1058, 16, 24, 32, 82, 104findcard2s 7094 . 2  |-  ( A  e.  Fin  ->  A. k  e.  ZZ  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) )
106 oveq2 5827 . . . 4  |-  ( k  =  K  ->  (
( # `  A )  _C  k )  =  ( ( # `  A
)  _C  K ) )
107 eqeq2 2292 . . . . . 6  |-  ( k  =  K  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  K ) )
108107rabbidv 2780 . . . . 5  |-  ( k  =  K  ->  { x  e.  ~P A  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  K }
)
109108fveq2d 5489 . . . 4  |-  ( k  =  K  ->  ( # `
 { x  e. 
~P A  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
110106, 109eqeq12d 2297 . . 3  |-  ( k  =  K  ->  (
( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  K
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  K }
) ) )
111110rspccva 2883 . 2  |-  ( ( A. k  e.  ZZ  ( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  /\  K  e.  ZZ )  ->  (
( # `  A )  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
112105, 111sylan 457 1  |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A
)  _C  K )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    u. cun 3150   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5220  (class class class)co 5819   Fincfn 6858   0cc0 8732   1c1 8733   NN0cn0 9960   ZZcz 10019   ...cfz 10777    _C cbc 11310   #chash 11332
This theorem is referenced by:  hashbc2  13048  sylow1lem1  14904  musum  20426  ballotlem1  23040  ballotlem2  23042
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 4186  ax-pr 4212  ax-un 4510  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
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 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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-fz 10778  df-seq 11042  df-fac 11284  df-bc 11311  df-hash 11333
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