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Theorem hashbcval 14954
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 3054 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 pwexg 4587 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
32adantr 467 . . . 4  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  ~P A  e.  _V )
4 rabexg 4553 . . . 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 5865 . . . . . . 7  |-  ( b  =  x  ->  ( # `
 b )  =  ( # `  x
) )
76eqeq1d 2453 . . . . . 6  |-  ( b  =  x  ->  (
( # `  b )  =  i  <->  ( # `  x
)  =  i ) )
87cbvrabv 3044 . . . . 5  |-  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P a  |  (
# `  x )  =  i }
9 simpl 459 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  a  =  A )
109pweqd 3956 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ~P a  =  ~P A )
11 simpr 463 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  i  =  N )
1211eqeq2d 2461 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ( ( # `  x
)  =  i  <->  ( # `  x
)  =  N ) )
1310, 12rabeqbidv 3040 . . . . 5  |-  ( ( a  =  A  /\  i  =  N )  ->  { x  e.  ~P a  |  ( # `  x
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
148, 13syl5eq 2497 . . . 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 6426 . . 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 1365 . 2  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
181, 17sylan 474 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 371    = wceq 1444    e. wcel 1887   {crab 2741   _Vcvv 3045   ~Pcpw 3951   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   NN0cn0 10869   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295
This theorem is referenced by:  hashbccl  14955  hashbcss  14956  hashbc0  14957  hashbc2  14958  ramval  14960  ramvalOLD  14961  ram0  14980  ramub1lem1  14984  ramub1lem2  14985
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