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Theorem iinxprg 3979
Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
iinxprg.1  |-  ( x  =  A  ->  C  =  D )
iinxprg.2  |-  ( x  =  B  ->  C  =  E )
Assertion
Ref Expression
iinxprg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
|^|_ x  e.  { A ,  B } C  =  ( D  i^i  E
) )
Distinct variable groups:    x, A    x, B    x, D    x, E
Allowed substitution hints:    C( x)    V( x)    W( x)

Proof of Theorem iinxprg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iinxprg.1 . . . . 5  |-  ( x  =  A  ->  C  =  D )
21eleq2d 2350 . . . 4  |-  ( x  =  A  ->  (
y  e.  C  <->  y  e.  D ) )
3 iinxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
43eleq2d 2350 . . . 4  |-  ( x  =  B  ->  (
y  e.  C  <->  y  e.  E ) )
52, 4ralprg 3682 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } y  e.  C  <->  ( y  e.  D  /\  y  e.  E )
) )
65abbidv 2397 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { y  |  A. x  e.  { A ,  B } y  e.  C }  =  {
y  |  ( y  e.  D  /\  y  e.  E ) } )
7 df-iin 3908 . 2  |-  |^|_ x  e.  { A ,  B } C  =  {
y  |  A. x  e.  { A ,  B } y  e.  C }
8 df-in 3159 . 2  |-  ( D  i^i  E )  =  { y  |  ( y  e.  D  /\  y  e.  E ) }
96, 7, 83eqtr4g 2340 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
|^|_ x  e.  { A ,  B } C  =  ( D  i^i  E
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    i^i cin 3151   {cpr 3641   |^|_ciin 3906
This theorem is referenced by:  pmapmeet  29962  diameetN  31246  dihmeetlem2N  31489  dihmeetcN  31492  dihmeet  31533
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-3an 936  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-ral 2548  df-v 2790  df-sbc 2992  df-un 3157  df-in 3159  df-sn 3646  df-pr 3647  df-iin 3908
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