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Theorem hashbcss 14606
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 995 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  C_  A )
2 sspwb 4686 . . . 4  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2sylib 196 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ~P B  C_  ~P A )
4 rabss2 3569 . . 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 994 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  A  e.  V )
76, 1ssexd 4584 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  e.  _V )
8 simp3 996 . . 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 14604 . . 3  |-  ( ( B  e.  _V  /\  N  e.  NN0 )  -> 
( B C N )  =  { x  e.  ~P B  |  (
# `  x )  =  N } )
117, 8, 10syl2anc 659 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  =  { x  e.  ~P B  |  ( # `  x
)  =  N }
)
129hashbcval 14604 . . 3  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
13123adant2 1013 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x
)  =  N }
)
145, 11, 133sstr4d 3532 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  C_  ( A C N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   NN0cn0 10791   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  ramval  14610  ramub2  14616  ramub1lem2  14629
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