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Theorem iinxprg 3995
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 2363 . . . 4  |-  ( x  =  A  ->  (
y  e.  C  <->  y  e.  D ) )
3 iinxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
43eleq2d 2363 . . . 4  |-  ( x  =  B  ->  (
y  e.  C  <->  y  e.  E ) )
52, 4ralprg 3695 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } y  e.  C  <->  ( y  e.  D  /\  y  e.  E )
) )
65abbidv 2410 . 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 3924 . 2  |-  |^|_ x  e.  { A ,  B } C  =  {
y  |  A. x  e.  { A ,  B } y  e.  C }
8 df-in 3172 . 2  |-  ( D  i^i  E )  =  { y  |  ( y  e.  D  /\  y  e.  E ) }
96, 7, 83eqtr4g 2353 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 1632    e. wcel 1696   {cab 2282   A.wral 2556    i^i cin 3164   {cpr 3654   |^|_ciin 3922
This theorem is referenced by:  pmapmeet  30584  diameetN  31868  dihmeetlem2N  32111  dihmeetcN  32114  dihmeet  32155
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-3an 936  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-ral 2561  df-v 2803  df-sbc 3005  df-un 3170  df-in 3172  df-sn 3659  df-pr 3660  df-iin 3924
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