Theorem cbvixp 6539

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
cbvixp.1	|- F/_ y B
cbvixp.2	|- F/_ x C
cbvixp.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbvixp	|- X_ x e. A B = X_ y e. A C

Distinct variable group:   x, A, y

Allowed substitution hints:   B( x, y)   C( x, y)

Proof of Theorem cbvixp

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvixp.1	. . . . . 6 |- F/_ y B
2	1	nfel2 2253	. . . . 5 |- F/ y ( f ` x ) e. B
3		cbvixp.2	. . . . . 6 |- F/_ x C
4	3	nfel2 2253	. . . . 5 |- F/ x ( f ` y ) e. C
5		fveq2 5353	. . . . . 6 |- ( x = y -> ( f ` x ) = ( f ` y ) )
6		cbvixp.3	. . . . . 6 |- ( x = y -> B = C )
7	5, 6	eleq12d 2170	. . . . 5 |- ( x = y -> ( ( f ` x ) e. B <-> ( f ` y ) e. C ) )
8	2, 4, 7	cbvral 2608	. . . 4 |- ( A. x e. A ( f ` x ) e. B <-> A. y e. A ( f ` y ) e. C )
9	8	anbi2i 448	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) e. B ) <-> ( f Fn A /\ A. y e. A ( f ` y ) e. C ) )
10	9	abbii 2215	. 2 |- { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) } = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
11		dfixp 6524	. 2 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
12		dfixp 6524	. 2 |- X_ y e. A C = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
13	10, 11, 12	3eqtr4i 2130	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   {cab 2086   F/_wnfc 2227   A.wral 2375   Fn wfn 5054   ` cfv 5059   X_cixp 6522

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fn 5062  df-fv 5067  df-ixp 6523

This theorem is referenced by:  cbvixpv  6540  mptelixpg  6558