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Theorem riinm 3757
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 3157 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2m 3337 . . . . 5  |-  ( ( E. x  x  e.  X  /\  A. x  e.  X  S  C_  A
)  ->  E. x  e.  X  S  C_  A
)
32ancoms 259 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  E. x  e.  X  S  C_  A
)
4 iinss 3736 . . . 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 2959 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 131 . 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, 7syl5eq 2100 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 101    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324    i^i cin 2944    C_ wss 2945   |^|_ciin 3686
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-iin 3688
This theorem is referenced by:  riinerm  6210
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