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Theorem iinconstm 3880
Description: Indexed intersection of a constant class, i.e. where  B does not depend on  x. (Contributed by Jim Kingdon, 19-Dec-2018.)
Assertion
Ref Expression
iinconstm  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  B  =  B )
Distinct variable groups:    x, A    x, B    y, A
Allowed substitution hint:    B( y)

Proof of Theorem iinconstm
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 2733 . . . 4  |-  z  e. 
_V
2 eliin 3876 . . . 4  |-  ( z  e.  _V  ->  (
z  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  z  e.  B ) )
31, 2ax-mp 5 . . 3  |-  ( z  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  z  e.  B )
4 r19.3rmv 3504 . . 3  |-  ( E. y  y  e.  A  ->  ( z  e.  B  <->  A. x  e.  A  z  e.  B ) )
53, 4bitr4id 198 . 2  |-  ( E. y  y  e.  A  ->  ( z  e.  |^|_ x  e.  A  B  <->  z  e.  B ) )
65eqrdv 2168 1  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   A.wral 2448   _Vcvv 2730   |^|_ciin 3872
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-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-iin 3874
This theorem is referenced by:  iin0imm  4152
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