ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfixp Unicode version

Theorem dfixp 6601
Description: Eliminate the expression  { x  |  x  e.  A } in df-ixp 6600, 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 6600 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 abid2 2261 . . . . 5  |-  { x  |  x  e.  A }  =  A
32fneq2i 5225 . . . 4  |-  ( f  Fn  { x  |  x  e.  A }  <->  f  Fn  A )
43anbi1i 454 . . 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 2256 . 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 2161 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 103    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417    Fn wfn 5125   ` cfv 5130   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-fn 5133  df-ixp 6600
This theorem is referenced by:  ixpsnval  6602  elixp2  6603  ixpeq1  6610  cbvixp  6616  ixp0x  6627
  Copyright terms: Public domain W3C validator