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Theorem iinxprg 3945
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 2240 . . . 4  |-  ( x  =  A  ->  (
y  e.  C  <->  y  e.  D ) )
3 iinxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
43eleq2d 2240 . . . 4  |-  ( x  =  B  ->  (
y  e.  C  <->  y  e.  E ) )
52, 4ralprg 3632 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } y  e.  C  <->  ( y  e.  D  /\  y  e.  E )
) )
65abbidv 2288 . 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 3874 . 2  |-  |^|_ x  e.  { A ,  B } C  =  {
y  |  A. x  e.  { A ,  B } y  e.  C }
8 df-in 3127 . 2  |-  ( D  i^i  E )  =  { y  |  ( y  e.  D  /\  y  e.  E ) }
96, 7, 83eqtr4g 2228 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 103    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448    i^i cin 3120   {cpr 3582   |^|_ciin 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-sn 3587  df-pr 3588  df-iin 3874
This theorem is referenced by: (None)
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