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Theorem watfvalN 33633
Description: The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
watomfval.a  |-  A  =  ( Atoms `  K )
watomfval.p  |-  P  =  ( _|_P `  K )
watomfval.w  |-  W  =  ( WAtoms `  K )
Assertion
Ref Expression
watfvalN  |-  ( K  e.  B  ->  W  =  ( d  e.  A  |->  ( A  \ 
( ( _|_P `  K ) `  {
d } ) ) ) )
Distinct variable groups:    A, d    K, d
Allowed substitution hints:    B( d)    P( d)    W( d)

Proof of Theorem watfvalN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2979 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 watomfval.w . . 3  |-  W  =  ( WAtoms `  K )
3 fveq2 5689 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 watomfval.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2491 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5689 . . . . . . 7  |-  ( k  =  K  ->  ( _|_P `  k )  =  ( _|_P `  K ) )
76fveq1d 5691 . . . . . 6  |-  ( k  =  K  ->  (
( _|_P `  k ) `  {
d } )  =  ( ( _|_P `  K ) `  {
d } ) )
85, 7difeq12d 3473 . . . . 5  |-  ( k  =  K  ->  (
( Atoms `  k )  \  ( ( _|_P `  k ) `
 { d } ) )  =  ( A  \  ( ( _|_P `  K
) `  { d } ) ) )
95, 8mpteq12dv 4368 . . . 4  |-  ( k  =  K  ->  (
d  e.  ( Atoms `  k )  |->  ( (
Atoms `  k )  \ 
( ( _|_P `  k ) `  {
d } ) ) )  =  ( d  e.  A  |->  ( A 
\  ( ( _|_P `  K ) `
 { d } ) ) ) )
10 df-watsN 33631 . . . 4  |-  WAtoms  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  ( (
Atoms `  k )  \ 
( ( _|_P `  k ) `  {
d } ) ) ) )
11 fvex 5699 . . . . . 6  |-  ( Atoms `  K )  e.  _V
124, 11eqeltri 2511 . . . . 5  |-  A  e. 
_V
1312mptex 5946 . . . 4  |-  ( d  e.  A  |->  ( A 
\  ( ( _|_P `  K ) `
 { d } ) ) )  e. 
_V
149, 10, 13fvmpt 5772 . . 3  |-  ( K  e.  _V  ->  ( WAtoms `
 K )  =  ( d  e.  A  |->  ( A  \  (
( _|_P `  K ) `  {
d } ) ) ) )
152, 14syl5eq 2485 . 2  |-  ( K  e.  _V  ->  W  =  ( d  e.  A  |->  ( A  \ 
( ( _|_P `  K ) `  {
d } ) ) ) )
161, 15syl 16 1  |-  ( K  e.  B  ->  W  =  ( d  e.  A  |->  ( A  \ 
( ( _|_P `  K ) `  {
d } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2970    \ cdif 3323   {csn 3875    e. cmpt 4348   ` cfv 5416   Atomscatm 32905   _|_PcpolN 33543   WAtomscwpointsN 33627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-watsN 33631
This theorem is referenced by:  watvalN  33634
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