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Theorem iinconstm 3747
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 r19.3rmv 3378 . . 3  |-  ( E. y  y  e.  A  ->  ( z  e.  B  <->  A. x  e.  A  z  e.  B ) )
2 vex 2625 . . . 4  |-  z  e. 
_V
3 eliin 3743 . . . 4  |-  ( z  e.  _V  ->  (
z  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  z  e.  B ) )
42, 3ax-mp 7 . . 3  |-  ( z  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  z  e.  B )
51, 4syl6rbbr 198 . 2  |-  ( E. y  y  e.  A  ->  ( z  e.  |^|_ x  e.  A  B  <->  z  e.  B ) )
65eqrdv 2087 1  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290   E.wex 1427    e. wcel 1439   A.wral 2360   _Vcvv 2622   |^|_ciin 3739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-iin 3741
This theorem is referenced by:  iin0imm  4011
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