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Theorem watfvalN 35817
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 3118 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 watomfval.w . . 3  |-  W  =  ( WAtoms `  K )
3 fveq2 5872 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 watomfval.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2516 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( _|_P `  k )  =  ( _|_P `  K ) )
76fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( _|_P `  k ) `  {
d } )  =  ( ( _|_P `  K ) `  {
d } ) )
85, 7difeq12d 3619 . . . . 5  |-  ( k  =  K  ->  (
( Atoms `  k )  \  ( ( _|_P `  k ) `
 { d } ) )  =  ( A  \  ( ( _|_P `  K
) `  { d } ) ) )
95, 8mpteq12dv 4535 . . . 4  |-  ( k  =  K  ->  (
d  e.  ( Atoms `  k )  |->  ( (
Atoms `  k )  \ 
( ( _|_P `  k ) `  {
d } ) ) )  =  ( d  e.  A  |->  ( A 
\  ( ( _|_P `  K ) `
 { d } ) ) ) )
10 df-watsN 35815 . . . 4  |-  WAtoms  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  ( (
Atoms `  k )  \ 
( ( _|_P `  k ) `  {
d } ) ) ) )
11 fvex 5882 . . . . . 6  |-  ( Atoms `  K )  e.  _V
124, 11eqeltri 2541 . . . . 5  |-  A  e. 
_V
1312mptex 6144 . . . 4  |-  ( d  e.  A  |->  ( A 
\  ( ( _|_P `  K ) `
 { d } ) ) )  e. 
_V
149, 10, 13fvmpt 5956 . . 3  |-  ( K  e.  _V  ->  ( WAtoms `
 K )  =  ( d  e.  A  |->  ( A  \  (
( _|_P `  K ) `  {
d } ) ) ) )
152, 14syl5eq 2510 . 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 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468   {csn 4032    |-> cmpt 4515   ` cfv 5594   Atomscatm 35089   _|_PcpolN 35727   WAtomscwpointsN 35811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-watsN 35815
This theorem is referenced by:  watvalN  35818
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