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Theorem dfixp 7471
Description: Eliminate the expression  { x  |  x  e.  A } in df-ixp 7470, 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 7470 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 abid2 2607 . . . . 5  |-  { x  |  x  e.  A }  =  A
32fneq2i 5676 . . . 4  |-  ( f  Fn  { x  |  x  e.  A }  <->  f  Fn  A )
43anbi1i 695 . . 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 2601 . 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 2496 1  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814    Fn wfn 5583   ` cfv 5588   X_cixp 7469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-fn 5591  df-ixp 7470
This theorem is referenced by:  ixpsnval  7472  elixp2  7473  ixpeq1  7480  cbvixp  7486  ixp0x  7497
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