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Theorem hashbcval 14061
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 2979 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 pwexg 4474 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
32adantr 465 . . . 4  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  ~P A  e.  _V )
4 rabexg 4440 . . . 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 5689 . . . . . . 7  |-  ( b  =  x  ->  ( # `
 b )  =  ( # `  x
) )
76eqeq1d 2449 . . . . . 6  |-  ( b  =  x  ->  (
( # `  b )  =  i  <->  ( # `  x
)  =  i ) )
87cbvrabv 2969 . . . . 5  |-  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P a  |  (
# `  x )  =  i }
9 simpl 457 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  a  =  A )
109pweqd 3863 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ~P a  =  ~P A )
11 simpr 461 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  i  =  N )
1211eqeq2d 2452 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ( ( # `  x
)  =  i  <->  ( # `  x
)  =  N ) )
1310, 12rabeqbidv 2965 . . . . 5  |-  ( ( a  =  A  /\  i  =  N )  ->  { x  e.  ~P a  |  ( # `  x
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
148, 13syl5eq 2485 . . . 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 6218 . . 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 1315 . 2  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
181, 17sylan 471 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 369    = wceq 1369    e. wcel 1756   {crab 2717   _Vcvv 2970   ~Pcpw 3858   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   NN0cn0 10577   #chash 12101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094
This theorem is referenced by:  hashbccl  14062  hashbcss  14063  hashbc0  14064  hashbc2  14065  ramval  14067  ram0  14081  ramub1lem1  14085  ramub1lem2  14086
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