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Theorem riinm 3729
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 3129 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2m 3309 . . . . 5  |-  ( ( E. x  x  e.  X  /\  A. x  e.  X  S  C_  A
)  ->  E. x  e.  X  S  C_  A
)
32ancoms 255 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  E. x  e.  X  S  C_  A
)
4 iinss 3708 . . . 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 2931 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 127 . 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 2084 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 97    = wceq 1243   E.wex 1381    e. wcel 1393   A.wral 2306   E.wrex 2307    i^i cin 2916    C_ wss 2917   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iin 3660
This theorem is referenced by:  riinerm  6179
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