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Theorem riinn0 3992
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 3374 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2z 3556 . . . . 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 3969 . . . 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 3179 . . 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 2340 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 1632    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   |^|_ciin 3922
This theorem is referenced by:  riinrab  3993  riiner  6748  mreriincl  13516  riinopn  16670  alexsublem  17754  fnemeet1  26418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-iin 3924
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