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Theorem riinerm 6586
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 6585 . 2  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
2 eleq1 2233 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
32cbvexv 1911 . . . . 5  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
4 eleq1 2233 . . . . . 6  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
54cbvexv 1911 . . . . 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 6536 . . . . . . 7  |-  ( R  Er  B  ->  R  C_  ( B  X.  B
) )
87ralimi 2533 . . . . . 6  |-  ( A. x  e.  A  R  Er  B  ->  A. x  e.  A  R  C_  ( B  X.  B ) )
9 riinm 3945 . . . . . 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 6520 . . . . 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 1348   E.wex 1485    e. wcel 2141   A.wral 2448    i^i cin 3120    C_ wss 3121   |^|_ciin 3874    X. cxp 4609    Er wer 6510
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iin 3876  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-er 6513
This theorem is referenced by: (None)
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