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Theorem hashbc 11407
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 5541 . . . . . 6  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
21oveq1d 5889 . . . . 5  |-  ( w  =  (/)  ->  ( (
# `  w )  _C  k )  =  ( ( # `  (/) )  _C  k ) )
3 pweq 3641 . . . . . . 7  |-  ( w  =  (/)  ->  ~P w  =  ~P (/) )
4 rabeq 2795 . . . . . . 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 5545 . . . . 5  |-  ( w  =  (/)  ->  ( # `  { x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) )
72, 6eqeq12d 2310 . . . 4  |-  ( w  =  (/)  ->  ( ( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } ) ) )
87ralbidv 2576 . . 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 5541 . . . . . 6  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
109oveq1d 5889 . . . . 5  |-  ( w  =  y  ->  (
( # `  w )  _C  k )  =  ( ( # `  y
)  _C  k ) )
11 pweq 3641 . . . . . . 7  |-  ( w  =  y  ->  ~P w  =  ~P y
)
12 rabeq 2795 . . . . . . 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 5545 . . . . 5  |-  ( w  =  y  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } ) )
1510, 14eqeq12d 2310 . . . 4  |-  ( w  =  y  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  k
)  =  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  k } ) ) )
1615ralbidv 2576 . . 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 5541 . . . . . 6  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  w
)  =  ( # `  ( y  u.  {
z } ) ) )
1817oveq1d 5889 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( ( # `  w )  _C  k
)  =  ( (
# `  ( y  u.  { z } ) )  _C  k ) )
19 pweq 3641 . . . . . . 7  |-  ( w  =  ( y  u. 
{ z } )  ->  ~P w  =  ~P ( y  u. 
{ z } ) )
20 rabeq 2795 . . . . . . 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 5545 . . . . 5  |-  ( w  =  ( y  u. 
{ z } )  ->  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  =  ( # `  { x  e.  ~P ( y  u.  {
z } )  |  ( # `  x
)  =  k } ) )
2318, 22eqeq12d 2310 . . . 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 2576 . . 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 5541 . . . . . 6  |-  ( w  =  A  ->  ( # `
 w )  =  ( # `  A
) )
2625oveq1d 5889 . . . . 5  |-  ( w  =  A  ->  (
( # `  w )  _C  k )  =  ( ( # `  A
)  _C  k ) )
27 pweq 3641 . . . . . . 7  |-  ( w  =  A  ->  ~P w  =  ~P A
)
28 rabeq 2795 . . . . . . 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 5545 . . . . 5  |-  ( w  =  A  ->  ( # `
 { x  e. 
~P w  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } ) )
3126, 30eqeq12d 2310 . . . 4  |-  ( w  =  A  ->  (
( ( # `  w
)  _C  k )  =  ( # `  {
x  e.  ~P w  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) ) )
3231ralbidv 2576 . . 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 11371 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
3433a1i 10 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 (/) )  =  0 )
35 elfz1eq 10823 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
3634, 35oveq12d 5892 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  ( 0  _C  0 ) )
37 0nn0 9996 . . . . . . . . 9  |-  0  e.  NN0
38 bcn0 11339 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0  _C  0 )  =  1 )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( 0  _C  0 )  =  1
4036, 39syl6eq 2344 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  (
( # `  (/) )  _C  k )  =  1 )
41 pw0 3778 . . . . . . . . . 10  |-  ~P (/)  =  { (/)
}
4235eqcomd 2301 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 0 )  ->  0  =  k )
4341raleqi 2753 . . . . . . . . . . . . 13  |-  ( A. x  e.  ~P  (/) ( # `  x )  =  k  <->  A. x  e.  { (/) }  ( # `  x
)  =  k )
44 0ex 4166 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
45 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
4645, 33syl6eq 2344 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
4746eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( (
# `  x )  =  k  <->  0  =  k ) )
4844, 47ralsn 3687 . . . . . . . . . . . . 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 2730 . . . . . . . . . . 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 2343 . . . . . . . . 9  |-  ( k  e.  ( 0 ... 0 )  ->  { x  e.  ~P (/)  |  ( # `
 x )  =  k }  =  { (/)
} )
5453fveq2d 5545 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  (
# `  { (/) } ) )
55 hashsng 11372 . . . . . . . . 9  |-  ( (/)  e.  _V  ->  ( # `  { (/)
} )  =  1 )
5644, 55ax-mp 8 . . . . . . . 8  |-  ( # `  { (/) } )  =  1
5754, 56syl6eq 2344 . . . . . . 7  |-  ( k  e.  ( 0 ... 0 )  ->  ( # `
 { x  e. 
~P (/)  |  ( # `  x )  =  k } )  =  1 )
5840, 57eqtr4d 2331 . . . . . 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 5884 . . . . . 6  |-  ( (
# `  (/) )  _C  k )  =  ( 0  _C  k )
61 bcval3 11335 . . . . . . . 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 10051 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
65 elfz3 10822 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
6664, 65ax-mp 8 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 ... 0
)
6763, 66syl6eqelr 2385 . . . . . . . . . . . . 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 2753 . . . . . . . . . . . 12  |-  ( A. x  e.  ~P  (/)  -.  ( # `
 x )  =  k  <->  A. x  e.  { (/)
}  -.  ( # `  x )  =  k )
7147notbid 285 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( -.  ( # `  x
)  =  k  <->  -.  0  =  k ) )
7244, 71ralsn 3687 . . . . . . . . . . . 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 3489 . . . . . . . . . 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 5545 . . . . . . . 8  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  ( # `  (/) ) )
7877, 33syl6eq 2344 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( # `  {
x  e.  ~P (/)  |  (
# `  x )  =  k } )  =  0 )
7962, 78eqtr4d 2331 . . . . . 6  |-  ( ( k  e.  ZZ  /\  -.  k  e.  (
0 ... 0 ) )  ->  ( 0  _C  k )  =  (
# `  { x  e.  ~P (/)  |  ( # `
 x )  =  k } ) )
8060, 79syl5eq 2340 . . . . 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 2621 . . 3  |-  A. k  e.  ZZ  ( ( # `  (/) )  _C  k
)  =  ( # `  { x  e.  ~P (/) 
|  ( # `  x
)  =  k } )
83 oveq2 5882 . . . . . 6  |-  ( k  =  j  ->  (
( # `  y )  _C  k )  =  ( ( # `  y
)  _C  j ) )
84 eqeq2 2305 . . . . . . . . 9  |-  ( k  =  j  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  j ) )
8584rabbidv 2793 . . . . . . . 8  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { x  e.  ~P y  |  ( # `  x
)  =  j } )
86 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  z  ->  ( # `
 x )  =  ( # `  z
) )
8786eqeq1d 2304 . . . . . . . . 9  |-  ( x  =  z  ->  (
( # `  x )  =  j  <->  ( # `  z
)  =  j ) )
8887cbvrabv 2800 . . . . . . . 8  |-  { x  e.  ~P y  |  (
# `  x )  =  j }  =  { z  e.  ~P y  |  ( # `  z
)  =  j }
8985, 88syl6eq 2344 . . . . . . 7  |-  ( k  =  j  ->  { x  e.  ~P y  |  (
# `  x )  =  k }  =  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9089fveq2d 5545 . . . . . 6  |-  ( k  =  j  ->  ( # `
 { x  e. 
~P y  |  (
# `  x )  =  k } )  =  ( # `  {
z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9183, 90eqeq12d 2310 . . . . 5  |-  ( k  =  j  ->  (
( ( # `  y
)  _C  k )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  k } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) ) )
9291cbvralv 2777 . . . 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 5544 . . . . . . . . . 10  |-  ( # `  { x  e.  ~P y  |  ( # `  x
)  =  j } )  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } )
9796eqeq2i 2306 . . . . . . . . 9  |-  ( ( ( # `  y
)  _C  j )  =  ( # `  {
x  e.  ~P y  |  ( # `  x
)  =  j } )  <->  ( ( # `  y )  _C  j
)  =  ( # `  { z  e.  ~P y  |  ( # `  z
)  =  j } ) )
9897ralbii 2580 . . . . . . . 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 11406 . . . . . 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 2646 . . . 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 7115 . 2  |-  ( A  e.  Fin  ->  A. k  e.  ZZ  ( ( # `  A )  _C  k
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  k } ) )
106 oveq2 5882 . . . 4  |-  ( k  =  K  ->  (
( # `  A )  _C  k )  =  ( ( # `  A
)  _C  K ) )
107 eqeq2 2305 . . . . . 6  |-  ( k  =  K  ->  (
( # `  x )  =  k  <->  ( # `  x
)  =  K ) )
108107rabbidv 2793 . . . . 5  |-  ( k  =  K  ->  { x  e.  ~P A  |  (
# `  x )  =  k }  =  { x  e.  ~P A  |  ( # `  x
)  =  K }
)
109108fveq2d 5545 . . . 4  |-  ( k  =  K  ->  ( # `
 { x  e. 
~P A  |  (
# `  x )  =  k } )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  K }
) )
110106, 109eqeq12d 2310 . . 3  |-  ( k  =  K  ->  (
( ( # `  A
)  _C  k )  =  ( # `  {
x  e.  ~P A  |  ( # `  x
)  =  k } )  <->  ( ( # `  A )  _C  K
)  =  ( # `  { x  e.  ~P A  |  ( # `  x
)  =  K }
) ) )
111110rspccva 2896 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    u. cun 3163   (/)c0 3468   ~Pcpw 3638   {csn 3653   ` cfv 5271  (class class class)co 5874   Fincfn 6879   0cc0 8753   1c1 8754   NN0cn0 9981   ZZcz 10040   ...cfz 10798    _C cbc 11331   #chash 11353
This theorem is referenced by:  hashbc2  13069  sylow1lem1  14925  musum  20447  ballotlem1  23061  ballotlem2  23063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332  df-hash 11354
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