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Theorem rintm 3813
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( A  i^i  |^| X )  = 
|^| X )
Distinct variable group:    x, X
Allowed substitution hint:    A( x)

Proof of Theorem rintm
StepHypRef Expression
1 incom 3190 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2m 3707 . . . 4  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_ 
U. ~P A )
3 ssid 3042 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 3808 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 143 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 3035 . . 3  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_  A )
7 df-ss 3010 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 120 . 2  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( |^| X  i^i  A )  =  |^| X )
91, 8syl5eq 2132 1  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( A  i^i  |^| X )  = 
|^| X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438    i^i cin 2996    C_ wss 2997   ~Pcpw 3425   U.cuni 3648   |^|cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-uni 3649  df-int 3684
This theorem is referenced by: (None)
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