Theorem dfixp 6810

Description: Eliminate the expression { x | x e. A } in df-ixp 6809, 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 6809	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2		abid2 2328	. . . . 5 |- { x | x e. A } = A
3	2	fneq2i 5388	. . . 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 2323	. 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 2228	1 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   Fn wfn 5285   ` cfv 5290   X_cixp 6808

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-fn 5293  df-ixp 6809

This theorem is referenced by:  ixpsnval  6811  elixp2  6812  ixpeq1  6819  cbvixp  6825  ixp0x  6836