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Theorem hashbcss 13364
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 4405 . . . 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 3418 . . 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 4342 . . 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 13362 . . 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 13362 . . 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 3383 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   NN0cn0 10213   #chash 11610
This theorem is referenced by:  ramval  13368  ramub2  13374  ramub1lem2  13387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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