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Theorem hashbcval 14906
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 3087 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 pwexg 4600 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
32adantr 466 . . . 4  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  ~P A  e.  _V )
4 rabexg 4566 . . . 4  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  ( # `  x
)  =  N }  e.  _V )
53, 4syl 17 . . 3  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  { x  e.  ~P A  |  ( # `  x
)  =  N }  e.  _V )
6 fveq2 5872 . . . . . . 7  |-  ( b  =  x  ->  ( # `
 b )  =  ( # `  x
) )
76eqeq1d 2422 . . . . . 6  |-  ( b  =  x  ->  (
( # `  b )  =  i  <->  ( # `  x
)  =  i ) )
87cbvrabv 3077 . . . . 5  |-  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P a  |  (
# `  x )  =  i }
9 simpl 458 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  a  =  A )
109pweqd 3981 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ~P a  =  ~P A )
11 simpr 462 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  i  =  N )
1211eqeq2d 2434 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ( ( # `  x
)  =  i  <->  ( # `  x
)  =  N ) )
1310, 12rabeqbidv 3073 . . . . 5  |-  ( ( a  =  A  /\  i  =  N )  ->  { x  e.  ~P a  |  ( # `  x
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
148, 13syl5eq 2473 . . . 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 6431 . . 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 1361 . 2  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
181, 17sylan 473 1  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   {crab 2777   _Vcvv 3078   ~Pcpw 3976   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   NN0cn0 10858   #chash 12501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  hashbccl  14907  hashbcss  14908  hashbc0  14909  hashbc2  14910  ramval  14912  ramvalOLD  14913  ram0  14932  ramub1lem1  14936  ramub1lem2  14937
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