Theorem dfixp 6972

Description: Eliminate the expression { x | x e. A } in df-ixp 6971, 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

Step	Hyp	Ref	Expression
1		df-ixp 6971	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2		abid2 2361	. . . . 5 |- { x | x e. A } = A
3	2	fneq2i 5471	. . . 4 |- ( f Fn { x | x e. A } <-> f Fn A )
4	3	anbi1i 462	. . 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 ) )
5	4	abbii 2354	. 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 ) }
6	1, 5	eqtri 2259	1 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   Fn wfn 5367   ` cfv 5372   X_cixp 6970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-fn 5375  df-ixp 6971

This theorem is referenced by:  ixpsnval  6973  elixp2  6974  ixpeq1  6981  cbvixp  6987  ixp0x  6998