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Theorem riiner 6978
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 6976 . . 3  |-  ( B  X.  B )  Er  B
2 riin0 4165 . . . . 5  |-  ( A  =  (/)  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
32adantl 454 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
4 ereq1 6913 . . . 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 16 . . 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 226 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
7 iiner 6977 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
87ancoms 441 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  R  Er  B )
9 erssxp 6929 . . . . . 6  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
109ralimi 2782 . . . . 5  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
11 riinn0 4166 . . . . 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 459 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  = 
|^|_ x  e.  A  R )
13 ereq1 6913 . . . 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 16 . . 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 225 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B )
166, 15pm2.61dane 2683 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 178    /\ wa 360    = wceq 1653    =/= wne 2600   A.wral 2706    i^i cin 3320    C_ wss 3321   (/)c0 3629   |^|_ciin 4095    X. cxp 4877    Er wer 6903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-iin 4097  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-er 6906
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