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Theorem iinconstm 3694
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 3340 . . 3  |-  ( E. y  y  e.  A  ->  ( z  e.  B  <->  A. x  e.  A  z  e.  B ) )
2 vex 2577 . . . 4  |-  z  e. 
_V
3 eliin 3690 . . . 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 192 . 2  |-  ( E. y  y  e.  A  ->  ( z  e.  |^|_ x  e.  A  B  <->  z  e.  B ) )
65eqrdv 2054 1  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   _Vcvv 2574   |^|_ciin 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-iin 3688
This theorem is referenced by:  iin0imm  3949
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