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Theorem dfixp 6835
Description: Eliminate the expression  { x  |  x  e.  A } in df-ixp 6834, 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 6834 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 abid2 2413 . . . . 5  |-  { x  |  x  e.  A }  =  A
32fneq2i 5355 . . . 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 2408 . 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 2316 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 1632    e. wcel 1696   {cab 2282   A.wral 2556    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  elixp2  6836  ixpeq1  6843  cbvixp  6849  ixp0x  6860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-fn 5274  df-ixp 6834
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