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Theorem iinconst 3930
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 3560 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  A. x  e.  A  y  e.  B )
)
2 vex 2804 . . . 4  |-  y  e. 
_V
3 eliin 3926 . . . 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 2294 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801   (/)c0 3468   |^|_ciin 3922
This theorem is referenced by:  iin0  4200  ptbasfi  17292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-v 2803  df-dif 3168  df-nul 3469  df-iin 3924
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