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Theorem hashbcval 13333
Description: Value of the "binomial set", the set of all  N-element subsets of  A. (Contributed by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
ramval.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
Assertion
Ref Expression
hashbcval  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
Distinct variable groups:    x, C    a, b, i, x    A, a, i, x    N, a, i, x    x, V
Allowed substitution hints:    A( b)    C( i, a, b)    N( b)    V( i, a, b)

Proof of Theorem hashbcval
StepHypRef Expression
1 elex 2932 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 pwexg 4351 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
32adantr 452 . . . 4  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  ~P A  e.  _V )
4 rabexg 4321 . . . 4  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  ( # `  x
)  =  N }  e.  _V )
53, 4syl 16 . . 3  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  { x  e.  ~P A  |  ( # `  x
)  =  N }  e.  _V )
6 fveq2 5695 . . . . . . 7  |-  ( b  =  x  ->  ( # `
 b )  =  ( # `  x
) )
76eqeq1d 2420 . . . . . 6  |-  ( b  =  x  ->  (
( # `  b )  =  i  <->  ( # `  x
)  =  i ) )
87cbvrabv 2923 . . . . 5  |-  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P a  |  (
# `  x )  =  i }
9 simpl 444 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  a  =  A )
109pweqd 3772 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ~P a  =  ~P A )
11 simpr 448 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  i  =  N )
1211eqeq2d 2423 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ( ( # `  x
)  =  i  <->  ( # `  x
)  =  N ) )
1310, 12rabeqbidv 2919 . . . . 5  |-  ( ( a  =  A  /\  i  =  N )  ->  { x  e.  ~P a  |  ( # `  x
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
148, 13syl5eq 2456 . . . 4  |-  ( ( a  =  A  /\  i  =  N )  ->  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
15 ramval.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
1614, 15ovmpt2ga 6170 . . 3  |-  ( ( A  e.  _V  /\  N  e.  NN0  /\  {
x  e.  ~P A  |  ( # `  x
)  =  N }  e.  _V )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x
)  =  N }
)
175, 16mpd3an3 1280 . 2  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
181, 17sylan 458 1  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924   ~Pcpw 3767   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   NN0cn0 10185   #chash 11581
This theorem is referenced by:  hashbccl  13334  hashbcss  13335  hashbc0  13336  hashbc2  13337  ramval  13339  ram0  13353  ramub1lem1  13357  ramub1lem2  13358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053
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