Theorem cbvixp 6681

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 2321	. . . . 5 |- F/ y ( f ` x ) e. B
3		cbvixp.2	. . . . . 6 |- F/_ x C
4	3	nfel2 2321	. . . . 5 |- F/ x ( f ` y ) e. C
5		fveq2 5486	. . . . . 6 |- ( x = y -> ( f ` x ) = ( f ` y ) )
6		cbvixp.3	. . . . . 6 |- ( x = y -> B = C )
7	5, 6	eleq12d 2237	. . . . 5 |- ( x = y -> ( ( f ` x ) e. B <-> ( f ` y ) e. C ) )
8	2, 4, 7	cbvral 2688	. . . 4 |- ( A. x e. A ( f ` x ) e. B <-> A. y e. A ( f ` y ) e. C )
9	8	anbi2i 453	. . 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 2282	. 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 6666	. 2 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
12		dfixp 6666	. 2 |- X_ y e. A C = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
13	10, 11, 12	3eqtr4i 2196	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   F/_wnfc 2295   A.wral 2444   Fn wfn 5183   ` cfv 5188   X_cixp 6664

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fn 5191  df-fv 5196  df-ixp 6665

This theorem is referenced by:  cbvixpv  6682  mptelixpg  6700