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Theorem iinconst 4094
Description: Indexed intersection of a constant class, i.e. where  B does not depend on  x. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
iinconst  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iinconst
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.3rzv 3713 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  A. x  e.  A  y  e.  B )
)
2 vex 2951 . . . 4  |-  y  e. 
_V
3 eliin 4090 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
42, 3ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
51, 4syl6rbbr 256 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  B 
<->  y  e.  B ) )
65eqrdv 2433 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948   (/)c0 3620   |^|_ciin 4086
This theorem is referenced by:  iin0  4365  ptbasfi  17605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621  df-iin 4088
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