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Theorem iinxprg 3727
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 2107 . . . 4  |-  ( x  =  A  ->  (
y  e.  C  <->  y  e.  D ) )
3 iinxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
43eleq2d 2107 . . . 4  |-  ( x  =  B  ->  (
y  e.  C  <->  y  e.  E ) )
52, 4ralprg 3417 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } y  e.  C  <->  ( y  e.  D  /\  y  e.  E )
) )
65abbidv 2155 . 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 3656 . 2  |-  |^|_ x  e.  { A ,  B } C  =  {
y  |  A. x  e.  { A ,  B } y  e.  C }
8 df-in 2921 . 2  |-  ( D  i^i  E )  =  { y  |  ( y  e.  D  /\  y  e.  E ) }
96, 7, 83eqtr4g 2097 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 97    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303    i^i cin 2913   {cpr 3373   |^|_ciin 3654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-sn 3378  df-pr 3379  df-iin 3656
This theorem is referenced by: (None)
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