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Theorem riinn0 3976
Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinn0  |-  ( ( A. x  e.  X  S  C_  A  /\  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 riinn0
StepHypRef Expression
1 incom 3361 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2z 3543 . . . . 5  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  S  C_  A )  ->  E. x  e.  X  S  C_  A
)
32ancoms 439 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  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 15 . . 3  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  |^|_ x  e.  X  S  C_  A
)
6 df-ss 3166 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 188 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7syl5eq 2327 1  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  riinrab  3977  riiner  6732  mreriincl  13500  riinopn  16654  alexsublem  17738  fnemeet1  26315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iin 3908
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