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Theorem riinerm 6553
Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
riinerm  |-  ( ( E. y  y  e.  A  /\  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    y, A
Allowed substitution hints:    B( y)    R( x, y)

Proof of Theorem riinerm
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 iinerm 6552 . 2  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
2 eleq1 2220 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
32cbvexv 1898 . . . . 5  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
4 eleq1 2220 . . . . . 6  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
54cbvexv 1898 . . . . 5  |-  ( E. a  a  e.  A  <->  E. y  y  e.  A
)
63, 5bitri 183 . . . 4  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
7 erssxp 6503 . . . . . . 7  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
87ralimi 2520 . . . . . 6  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
9 riinm 3921 . . . . . 6  |-  ( ( A. x  e.  A  R  C_  ( B  X.  B )  /\  E. x  x  e.  A
)  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R
)  =  |^|_ x  e.  A  R )
108, 9sylan 281 . . . . 5  |-  ( ( A. x  e.  A  R  Er  B  /\  E. x  x  e.  A
)  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R
)  =  |^|_ x  e.  A  R )
11 ereq1 6487 . . . . 5  |-  ( ( ( 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 ) )
1210, 11syl 14 . . . 4  |-  ( ( A. x  e.  A  R  Er  B  /\  E. x  x  e.  A
)  ->  ( (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B  <->  |^|_ x  e.  A  R  Er  B )
)
136, 12sylan2br 286 . . 3  |-  ( ( A. x  e.  A  R  Er  B  /\  E. y  y  e.  A
)  ->  ( (
( B  X.  B
)  i^i  |^|_ x  e.  A  R )  Er  B  <->  |^|_ x  e.  A  R  Er  B )
)
1413ancoms 266 . 2  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  (
( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B  <->  |^|_ x  e.  A  R  Er  B
) )
151, 14mpbird 166 1  |-  ( ( E. y  y  e.  A  /\  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    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435    i^i cin 3101    C_ wss 3102   |^|_ciin 3850    X. cxp 4584    Er wer 6477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-iin 3852  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-er 6480
This theorem is referenced by: (None)
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