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Theorem hashbcval 14531
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 3118 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 pwexg 4640 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
32adantr 465 . . . 4  |-  ( ( A  e.  _V  /\  N  e.  NN0 )  ->  ~P A  e.  _V )
4 rabexg 4606 . . . 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 5872 . . . . . . 7  |-  ( b  =  x  ->  ( # `
 b )  =  ( # `  x
) )
76eqeq1d 2459 . . . . . 6  |-  ( b  =  x  ->  (
( # `  b )  =  i  <->  ( # `  x
)  =  i ) )
87cbvrabv 3108 . . . . 5  |-  { b  e.  ~P a  |  ( # `  b
)  =  i }  =  { x  e. 
~P a  |  (
# `  x )  =  i }
9 simpl 457 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  a  =  A )
109pweqd 4020 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ~P a  =  ~P A )
11 simpr 461 . . . . . . 7  |-  ( ( a  =  A  /\  i  =  N )  ->  i  =  N )
1211eqeq2d 2471 . . . . . 6  |-  ( ( a  =  A  /\  i  =  N )  ->  ( ( # `  x
)  =  i  <->  ( # `  x
)  =  N ) )
1310, 12rabeqbidv 3104 . . . . 5  |-  ( ( a  =  A  /\  i  =  N )  ->  { x  e.  ~P a  |  ( # `  x
)  =  i }  =  { x  e. 
~P A  |  (
# `  x )  =  N } )
148, 13syl5eq 2510 . . . 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 1325 . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   ~Pcpw 4015   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   NN0cn0 10816   #chash 12407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  hashbccl  14532  hashbcss  14533  hashbc0  14534  hashbc2  14535  ramval  14537  ram0  14551  ramub1lem1  14555  ramub1lem2  14556
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