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Theorem riiner 6748
Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
riiner  |-  ( A. x  e.  A  R  Er  B  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    R( x)

Proof of Theorem riiner
StepHypRef Expression
1 xpider 6746 . . 3  |-  ( B  X.  B )  Er  B
2 riin0 3991 . . . . 5  |-  ( A  =  (/)  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
32adantl 452 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
4 ereq1 6683 . . . 4  |-  ( ( ( B  X.  B
)  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B
)  ->  ( (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B  <->  ( B  X.  B )  Er  B
) )
53, 4syl 15 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( ( B  X.  B )  i^i  |^|_ x  e.  A  R
)  Er  B  <->  ( B  X.  B )  Er  B
) )
61, 5mpbiri 224 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
7 iiner 6747 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
87ancoms 439 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  R  Er  B )
9 erssxp 6699 . . . . . 6  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
109ralimi 2631 . . . . 5  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
11 riinn0 3992 . . . . 5  |-  ( ( A. x  e.  A  R  C_  ( B  X.  B )  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  = 
|^|_ x  e.  A  R )
1210, 11sylan 457 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  = 
|^|_ x  e.  A  R )
13 ereq1 6683 . . . 4  |-  ( ( ( B  X.  B
)  i^i  |^|_ x  e.  A  R )  = 
|^|_ x  e.  A  R  ->  ( ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B  <->  |^|_
x  e.  A  R  Er  B ) )
1412, 13syl 15 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B  <->  |^|_ x  e.  A  R  Er  B
) )
158, 14mpbird 223 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B )
166, 15pm2.61dane 2537 1  |-  ( A. x  e.  A  R  Er  B  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    =/= wne 2459   A.wral 2556    i^i cin 3164    C_ wss 3165   (/)c0 3468   |^|_ciin 3922    X. cxp 4703    Er wer 6673
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iin 3924  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-er 6676
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