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Theorem iinconst 4309
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 3892 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  A. x  e.  A  y  e.  B )
)
2 vex 3083 . . . 4  |-  y  e. 
_V
3 eliin 4305 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
42, 3ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
51, 4syl6rbbr 267 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  B 
<->  y  e.  B ) )
65eqrdv 2419 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   _Vcvv 3080   (/)c0 3761   |^|_ciin 4300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-v 3082  df-dif 3439  df-nul 3762  df-iin 4302
This theorem is referenced by:  iin0  4598  ptbasfi  20594
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