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Theorem iinconst 3914
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 3547 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  A. x  e.  A  y  e.  B )
)
2 vex 2791 . . . 4  |-  y  e. 
_V
3 eliin 3910 . . . 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 255 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  B 
<->  y  e.  B ) )
65eqrdv 2281 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  iin0  4184  ptbasfi  17276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456  df-iin 3908
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