Theorem cbvixp 6950

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 2397	. . . . 5 |- F/ y ( f ` x ) e. B
3		cbvixp.2	. . . . . 6 |- F/_ x C
4	3	nfel2 2397	. . . . 5 |- F/ x ( f ` y ) e. C
5		fveq2 5670	. . . . . 6 |- ( x = y -> ( f ` x ) = ( f ` y ) )
6		cbvixp.3	. . . . . 6 |- ( x = y -> B = C )
7	5, 6	eleq12d 2303	. . . . 5 |- ( x = y -> ( ( f ` x ) e. B <-> ( f ` y ) e. C ) )
8	2, 4, 7	cbvral 2774	. . . 4 |- ( A. x e. A ( f ` x ) e. B <-> A. y e. A ( f ` y ) e. C )
9	8	anbi2i 457	. . 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 2348	. 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 6935	. 2 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
12		dfixp 6935	. 2 |- X_ y e. A C = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
13	10, 11, 12	3eqtr4i 2263	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   F/_wnfc 2371   A.wral 2520   Fn wfn 5347   ` cfv 5352   X_cixp 6933

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fn 5355  df-fv 5360  df-ixp 6934

This theorem is referenced by:  cbvixpv  6951  mptelixpg  6969  prdsbas3  13500