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Theorem dfixp 6819
Description: Eliminate the expression  { x  |  x  e.  A } in df-ixp 6818, 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 6818 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 abid2 2400 . . . . 5  |-  { x  |  x  e.  A }  =  A
32fneq2i 5339 . . . 4  |-  ( f  Fn  { x  |  x  e.  A }  <->  f  Fn  A )
43anbi1i 676 . . 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 2395 . 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 2303 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 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  elixp2  6820  ixpeq1  6827  cbvixp  6833  ixp0x  6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-fn 5258  df-ixp 6818
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