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Theorem hashbcss 13300
Description: Subset relation for the binomial set. (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
hashbcss  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  C_  ( A C N ) )
Distinct variable groups:    a, b,
i    A, a, i    B, a, i    N, a, i
Allowed substitution hints:    A( b)    B( b)    C( i, a, b)    N( b)    V( i, a, b)

Proof of Theorem hashbcss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  C_  A )
2 sspwb 4355 . . . 4  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2sylib 189 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ~P B  C_  ~P A )
4 rabss2 3370 . . 3  |-  ( ~P B  C_  ~P A  ->  { x  e.  ~P B  |  ( # `  x
)  =  N }  C_ 
{ x  e.  ~P A  |  ( # `  x
)  =  N }
)
53, 4syl 16 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  { x  e.  ~P B  |  (
# `  x )  =  N }  C_  { x  e.  ~P A  |  (
# `  x )  =  N } )
6 simp1 957 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  A  e.  V )
76, 1ssexd 4292 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  e.  _V )
8 simp3 959 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  N  e.  NN0 )
9 ramval.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
109hashbcval 13298 . . 3  |-  ( ( B  e.  _V  /\  N  e.  NN0 )  -> 
( B C N )  =  { x  e.  ~P B  |  (
# `  x )  =  N } )
117, 8, 10syl2anc 643 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  =  { x  e.  ~P B  |  ( # `  x
)  =  N }
)
129hashbcval 13298 . . 3  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
13123adant2 976 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x
)  =  N }
)
145, 11, 133sstr4d 3335 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  C_  ( A C N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900    C_ wss 3264   ~Pcpw 3743   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   NN0cn0 10154   #chash 11546
This theorem is referenced by:  ramval  13304  ramub2  13310  ramub1lem2  13323
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026
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