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Theorem hashbc 11357
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
StepHypRef Expression
1 fveq2 5458 . . . . . 6  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
21oveq1d 5807 . . . . 5  |-  ( w  =  (/)  ->  ( (
# `  w )  _C  k )  =  ( ( # `  (/) )  _C  k ) )
3 pweq 3602 . . . . . . 7  |-  ( w  =  (/)  ->  ~P w  =  ~P (/) )
4 rabeq 2757 . . . . . . 7  |-  ( ~P w  =  ~P (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
53, 4syl 17 . . . . . 6  |-  ( w  =  (/)  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
65fveq2d 5462 . . . . 5  |-  ( w  =  (/)  ->  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
72, 6eqeq12d 2272 . . . 4  |-  ( w  =  (/)  ->  ( ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) ) )
87ralbidv 2538 . . 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 5458 . . . . . 6  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
109oveq1d 5807 . . . . 5  |-  ( w  =  y  ->  (
( # `  w )  _C  k )  =  ( ( # `  y
)  _C  k ) )
11 pweq 3602 . . . . . . 7  |-  ( w  =  y  ->  ~P w  =  ~P y
)
12 rabeq 2757 . . . . . . 7  |-  ( ~P w  =  ~P y  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P y  |  (
# `  x )  =  k } )
1311, 12syl 17 . . . . . 6  |-  ( w  =  y  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  k } )
1413fveq2d 5462 . . . . 5  |-  ( w  =  y  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) )
1510, 14eqeq12d 2272 . . . 4  |-  ( w  =  y  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  k
)  =  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
1615ralbidv 2538 . . 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 5458 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  w
)  =  ( # `  ( y  u.  {
z } ) ) )
1817oveq1d 5807 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( # `  w )  _C  k
)  =  ( (
# `  ( y  u.  { z } ) )  _C  k ) )
19 pweq 3602 . . . . . . 7  |-  ( w  =  ( y  u. 
{ z } )  ->  ~P w  =  ~P ( y  u. 
{ z } ) )
20 rabeq 2757 . . . . . . 7  |-  ( ~P w  =  ~P (
y  u.  { z } )  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2119, 20syl 17 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  { x  e. 
~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } )
2221fveq2d 5462 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } ) )
2318, 22eqeq12d 2272 . . . 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 2538 . . 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 5458 . . . . . 6  |-  ( w  =  A  ->  ( # `
 w )  =  ( # `  A
) )
2625oveq1d 5807 . . . . 5  |-  ( w  =  A  ->  (
( # `  w )  _C  k )  =  ( ( # `  A
)  _C  k ) )
27 pweq 3602 . . . . . . 7  |-  ( w  =  A  ->  ~P w  =  ~P A
)
28 rabeq 2757 . . . . . . 7  |-  ( ~P w  =  ~P A  ->  { x  e.  ~P w  |  ( # `  x
)  =  k }  =  { x  e. 
~P A  |  (
# `  x )  =  k } )
2927, 28syl 17 . . . . . 6  |-  ( w  =  A  ->  { x  e.  ~P w  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  k } )
3029fveq2d 5462 . . . . 5  |-  ( w  =  A  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) )
3126, 30eqeq12d 2272 . . . 4  |-  ( w  =  A  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
3231ralbidv 2538 . . 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 11322 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
3433a1i 12 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 (/) )  =  0 )
35 elfz1eq 10774 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
3634, 35oveq12d 5810 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  ( 0  _C  0 ) )
37 0nn0 9948 . . . . . . . . 9  |-  0  e.  NN0
38 bcn0 11290 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
3937, 38ax-mp 10 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
4036, 39syl6eq 2306 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  1 )
41 pw0 3736 . . . . . . . . . 10  |-  ~P (/)  =  { (/)
}
4235eqcomd 2263 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 0 )  ->  0  =  k )
4341raleqi 2715 . . . . . . . . . . . . 13  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  A. x  e.  { (/) }  ( # `  x
)  =  k )
44 0ex 4124 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
45 fveq2 5458 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
4645, 33syl6eq 2306 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
4746eqeq1d 2266 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( (
# `  x )  =  k  <->  0  =  k ) )
4844, 47ralsn 3648 . . . . . . . . . . . . 13  |-  ( A. x  e.  { (/) }  ( # `
 x )  =  k  <->  0  =  k )
4943, 48bitri 242 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  0  =  k )
5042, 49sylibr 205 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... 0 )  ->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
51 rabid2 2692 . . . . . . . . . . 11  |-  ( ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k }  <->  A. x  e.  ~P  (/) ( # `  x
)  =  k )
5250, 51sylibr 205 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... 0 )  ->  ~P (/)  =  { x  e. 
~P (/)  |  ( # `  x )  =  k } )
5341, 52syl5reqr 2305 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  { x  e.  ~P (/)  |  ( # `
 x )  =  k }  =  { (/)
} )
5453fveq2d 5462 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  (
# `  { (/) } ) )
55 hashsng 11323 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( # `  { (/)
} )  =  1 )
5644, 55ax-mp 10 . . . . . . . 8  |-  ( # `  { (/) } )  =  1
5754, 56syl6eq 2306 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  1 )
5840, 57eqtr4d 2293 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
5958adantl 454 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  e.  ( 0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
6033oveq1i 5802 . . . . . 6  |-  ( (
# `  (/) )  _C  k )  =  ( 0  _C  k )
61 bcval3 11286 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  k  e.  ZZ  /\  -.  k  e.  ( 0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
6237, 61mp3an1 1269 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  0 )
63 id 21 . . . . . . . . . . . . . 14  |-  ( 0  =  k  ->  0  =  k )
64 0z 10003 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
65 elfz3 10773 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
6664, 65ax-mp 10 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 ... 0
)
6763, 66syl6eqelr 2347 . . . . . . . . . . . . 13  |-  ( 0  =  k  ->  k  e.  ( 0 ... 0
) )
6867con3i 129 . . . . . . . . . . . 12  |-  ( -.  k  e.  ( 0 ... 0 )  ->  -.  0  =  k
)
6968adantl 454 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  -.  0  =  k )
7041raleqi 2715 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  A. x  e.  { (/)
}  -.  ( # `  x )  =  k )
7147notbid 287 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( -.  ( # `  x
)  =  k  <->  -.  0  =  k ) )
7244, 71ralsn 3648 . . . . . . . . . . . 12  |-  ( A. x  e.  { (/) }  -.  ( # `  x )  =  k  <->  -.  0  =  k )
7370, 72bitri 242 . . . . . . . . . . 11  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  -.  0  =  k )
7469, 73sylibr 205 . . . . . . . . . 10  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  A. x  e.  ~P  (/) 
-.  ( # `  x
)  =  k )
75 rabeq0 3451 . . . . . . . . . 10  |-  ( { x  e.  ~P (/)  |  (
# `  x )  =  k }  =  (/)  <->  A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k )
7674, 75sylibr 205 . . . . . . . . 9  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  { x  e. 
~P (/)  |  ( # `  x )  =  k }  =  (/) )
7776fveq2d 5462 . . . . . . . 8  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  ( # `  (/) ) )
7877, 33syl6eq 2306 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  0 )
7962, 78eqtr4d 2293 . . . . . 6  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8060, 79syl5eq 2302 . . . . 5  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
8159, 80pm2.61dan 769 . . . 4  |-  ( k  e.  ZZ  ->  (
( # `  (/) )  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8281rgen 2583 . . 3  |-  A. k  e.  ZZ  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
83 oveq2 5800 . . . . . 6  |-  ( k  =  j  ->  (
( # `  y )  _C  k )  =  ( ( # `  y
)  _C  j ) )
84 eqeq2 2267 . . . . . . . . 9  |-  ( k  =  j  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  j ) )
8584rabbidv 2755 . . . . . . . 8  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  j } )
86 fveq2 5458 . . . . . . . . . 10  |-  ( x  =  z  ->  ( # `
 x )  =  ( # `  z
) )
8786eqeq1d 2266 . . . . . . . . 9  |-  ( x  =  z  ->  (
( # `  x )  =  j  <->  ( # `  z
)  =  j ) )
8887cbvrabv 2762 . . . . . . . 8  |-  { x  e.  ~P y  |  (
# `  x )  =  j }  =  { z  e.  ~P y  |  ( # `  z
)  =  j }
8985, 88syl6eq 2306 . . . . . . 7  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9089fveq2d 5462 . . . . . 6  |-  ( k  =  j  ->  ( # `
 { x  e. 
~P y  |  (
# `  x )  =  k } )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9183, 90eqeq12d 2272 . . . . 5  |-  ( k  =  j  ->  (
( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )
9291cbvralv 2739 . . . 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 733 . . . . . . 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 734 . . . . . . 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 736 . . . . . . . 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 5461 . . . . . . . . . 10  |-  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  j } )  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9796eqeq2i 2268 . . . . . . . . 9  |-  ( ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9897ralbii 2542 . . . . . . . 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 205 . . . . . . 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 735 . . . . . . 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 11356 . . . . . 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 601 . . . . 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 2608 . . . 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 210 . . 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 7067 . 2  |-  ( A  e.  Fin  ->  A. k  e.  ZZ  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) )
106 oveq2 5800 . . . 4  |-  ( k  =  K  ->  (
( # `  A )  _C  k )  =  ( ( # `  A
)  _C  K ) )
107 eqeq2 2267 . . . . . 6  |-  ( k  =  K  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  K ) )
108107rabbidv 2755 . . . . 5  |-  ( k  =  K  ->  { x  e.  ~P A  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  K }
)
109108fveq2d 5462 . . . 4  |-  ( k  =  K  ->  ( # `
 { x  e. 
~P A  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
110106, 109eqeq12d 2272 . . 3  |-  ( k  =  K  ->  (
( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  K
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  K }
) ) )
111110rcla4cva 2858 . 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 459 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 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   {crab 2522   _Vcvv 2763    u. cun 3125   (/)c0 3430   ~Pcpw 3599   {csn 3614   ` cfv 4673  (class class class)co 5792   Fincfn 6831   0cc0 8705   1c1 8706   NN0cn0 9933   ZZcz 9992   ...cfz 10749    _C cbc 11282   #chash 11304
This theorem is referenced by:  hashbc2  13016  sylow1lem1  14872  musum  20394  ballotlem1  23007  ballotlem2  23009
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-fz 10750  df-seq 11014  df-fac 11256  df-bc 11283  df-hash 11305
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