Theorem dfixp 6678

Description: Eliminate the expression { x | x e. A } in df-ixp 6677, 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 6677	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2		abid2 2291	. . . . 5 |- { x | x e. A } = A
3	2	fneq2i 5293	. . . 4 |- ( f Fn { x | x e. A } <-> f Fn A )
4	3	anbi1i 455	. . 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 2286	. 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 2191	1 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   Fn wfn 5193   ` cfv 5198   X_cixp 6676

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-fn 5201  df-ixp 6677

This theorem is referenced by:  ixpsnval  6679  elixp2  6680  ixpeq1  6687  cbvixp  6693  ixp0x  6704