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Theorem dfixp 7057
Description: Eliminate the expression  { x  |  x  e.  A } in df-ixp 7056, under the assumption that  A and  x are disjoint. This way, we can say that  x is bound in  X_ x  e.  A B even if it appears free in  A. (Contributed by Mario Carneiro, 12-Aug-2016.)
Assertion
Ref Expression
dfixp  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
Distinct variable groups:    x, f, A    B, f    x, A
Allowed substitution hint:    B( x)

Proof of Theorem dfixp
StepHypRef Expression
1 df-ixp 7056 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 abid2 2552 . . . . 5  |-  { x  |  x  e.  A }  =  A
32fneq2i 5532 . . . 4  |-  ( f  Fn  { x  |  x  e.  A }  <->  f  Fn  A )
43anbi1i 677 . . 3  |-  ( ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
)  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
) )
54abbii 2547 . 2  |-  { f  |  ( f  Fn 
{ x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B ) }  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) }
61, 5eqtri 2455 1  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697    Fn wfn 5441   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  elixp2  7058  ixpeq1  7065  cbvixp  7071  ixp0x  7082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-fn 5449  df-ixp 7056
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