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Theorem rintm 3772
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 3157 . 2  |-  ( A  i^i  |^| X )  =  ( |^| X  i^i  A )
2 intssuni2m 3667 . . . 4  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_ 
U. ~P A )
3 ssid 2992 . . . . 5  |-  ~P A  C_ 
~P A
4 sspwuni 3767 . . . . 5  |-  ( ~P A  C_  ~P A  <->  U. ~P A  C_  A
)
53, 4mpbi 137 . . . 4  |-  U. ~P A  C_  A
62, 5syl6ss 2985 . . 3  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  |^| X  C_  A )
7 df-ss 2959 . . 3  |-  ( |^| X  C_  A  <->  ( |^| X  i^i  A )  = 
|^| X )
86, 7sylib 131 . 2  |-  ( ( X  C_  ~P A  /\  E. x  x  e.  X )  ->  ( |^| X  i^i  A )  =  |^| X )
91, 8syl5eq 2100 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 101    = wceq 1259   E.wex 1397    e. wcel 1409    i^i cin 2944    C_ wss 2945   ~Pcpw 3387   U.cuni 3608   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-uni 3609  df-int 3644
This theorem is referenced by: (None)
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