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Theorem iinxprg 4396
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 2524 . . . 4  |-  ( x  =  A  ->  (
y  e.  C  <->  y  e.  D ) )
3 iinxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
43eleq2d 2524 . . . 4  |-  ( x  =  B  ->  (
y  e.  C  <->  y  e.  E ) )
52, 4ralprg 4065 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } y  e.  C  <->  ( y  e.  D  /\  y  e.  E )
) )
65abbidv 2590 . 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 4318 . 2  |-  |^|_ x  e.  { A ,  B } C  =  {
y  |  A. x  e.  { A ,  B } y  e.  C }
8 df-in 3468 . 2  |-  ( D  i^i  E )  =  { y  |  ( y  e.  D  /\  y  e.  E ) }
96, 7, 83eqtr4g 2520 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
|^|_ x  e.  { A ,  B } C  =  ( D  i^i  E
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804    i^i cin 3460   {cpr 4018   |^|_ciin 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108  df-sbc 3325  df-un 3466  df-in 3468  df-sn 4017  df-pr 4019  df-iin 4318
This theorem is referenced by:  pmapmeet  35894  diameetN  37180  dihmeetlem2N  37423  dihmeetcN  37426  dihmeet  37467
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