Theorem dfixp 6768

Description: Eliminate the expression { x | x e. A } in df-ixp 6767, 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 6767	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2		abid2 2317	. . . . 5 |- { x | x e. A } = A
3	2	fneq2i 5354	. . . 4 |- ( f Fn { x | x e. A } <-> f Fn A )
4	3	anbi1i 458	. . 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 2312	. 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 2217	1 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   Fn wfn 5254   ` cfv 5259   X_cixp 6766

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-fn 5262  df-ixp 6767

This theorem is referenced by:  ixpsnval  6769  elixp2  6770  ixpeq1  6777  cbvixp  6783  ixp0x  6794