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Theorem riinerm 6345
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 6344 . 2  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
2 eleq1 2150 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
32cbvexv 1843 . . . . 5  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
4 eleq1 2150 . . . . . 6  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
54cbvexv 1843 . . . . 5  |-  ( E. a  a  e.  A  <->  E. y  y  e.  A
)
63, 5bitri 182 . . . 4  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
7 erssxp 6295 . . . . . . 7  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
87ralimi 2438 . . . . . 6  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
9 riinm 3797 . . . . . 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 277 . . . . 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 6279 . . . . 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 282 . . 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 264 . 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 165 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359    i^i cin 2996    C_ wss 2997   |^|_ciin 3726    X. cxp 4426    Er wer 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-iin 3728  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-er 6272
This theorem is referenced by: (None)
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