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Theorem hashbcss 13053
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 956 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  C_  A )
2 sspwb 4225 . . . 4  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2sylib 188 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ~P B  C_  ~P A )
4 rabss2 3258 . . 3  |-  ( ~P B  C_  ~P A  ->  { x  e.  ~P B  |  ( # `  x
)  =  N }  C_ 
{ x  e.  ~P A  |  ( # `  x
)  =  N }
)
53, 4syl 15 . 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 955 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  A  e.  V )
7 ssexg 4162 . . . 4  |-  ( ( B  C_  A  /\  A  e.  V )  ->  B  e.  _V )
81, 6, 7syl2anc 642 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  B  e.  _V )
9 simp3 957 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  N  e.  NN0 )
10 ramval.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
1110hashbcval 13051 . . 3  |-  ( ( B  e.  _V  /\  N  e.  NN0 )  -> 
( B C N )  =  { x  e.  ~P B  |  (
# `  x )  =  N } )
128, 9, 11syl2anc 642 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  =  { x  e.  ~P B  |  ( # `  x
)  =  N }
)
1310hashbcval 13051 . . 3  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( A C N )  =  { x  e.  ~P A  |  (
# `  x )  =  N } )
14133adant2 974 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x
)  =  N }
)
155, 12, 143sstr4d 3223 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 934    = wceq 1625    e. wcel 1686   {crab 2549   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   NN0cn0 9967   #chash 11339
This theorem is referenced by:  ramval  13057  ramub2  13063  ramub1lem2  13076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865
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