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Theorem riiner 7436
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 7434 . . 3  |-  ( B  X.  B )  Er  B
2 riin0 4367 . . . . 5  |-  ( A  =  (/)  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
32adantl 467 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  =  ( B  X.  B ) )
4 ereq1 7370 . . . 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 17 . . 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 236 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =  (/) )  -> 
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
7 iiner 7435 . . . 4  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
87ancoms 454 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  R  Er  B )
9 erssxp 7386 . . . . . 6  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
109ralimi 2816 . . . . 5  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
11 riinn0 4368 . . . . 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 473 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  = 
|^|_ x  e.  A  R )
13 ereq1 7370 . . . 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 17 . . 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 235 . 2  |-  ( ( A. x  e.  A  R  Er  B  /\  A  =/=  (/) )  ->  (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B )
166, 15pm2.61dane 2740 1  |-  ( A. x  e.  A  R  Er  B  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    =/= wne 2616   A.wral 2773    i^i cin 3432    C_ wss 3433   (/)c0 3758   |^|_ciin 4294    X. cxp 4844    Er wer 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-iin 4296  df-br 4418  df-opab 4477  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-er 7363
This theorem is referenced by: (None)
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