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Theorem riinm 3974
Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
riinm  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Distinct variable groups:    x, A    x, X
Allowed substitution hint:    S( x)

Proof of Theorem riinm
StepHypRef Expression
1 incom 3342 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2m 3524 . . . . 5  |-  ( ( E. x  x  e.  X  /\  A. x  e.  X  S  C_  A
)  ->  E. x  e.  X  S  C_  A
)
32ancoms 268 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  E. x  e.  X  S  C_  A
)
4 iinss 3953 . . . 4  |-  ( E. x  e.  X  S  C_  A  ->  |^|_ x  e.  X  S  C_  A
)
53, 4syl 14 . . 3  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  |^|_ x  e.  X  S  C_  A
)
6 df-ss 3157 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 122 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7eqtrid 2234 1  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469    i^i cin 3143    C_ wss 3144   |^|_ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-iin 3904
This theorem is referenced by:  riinerm  6634
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