Theorem cbvixp 6883

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 2387	. . . . 5 |- F/ y ( f ` x ) e. B
3		cbvixp.2	. . . . . 6 |- F/_ x C
4	3	nfel2 2387	. . . . 5 |- F/ x ( f ` y ) e. C
5		fveq2 5639	. . . . . 6 |- ( x = y -> ( f ` x ) = ( f ` y ) )
6		cbvixp.3	. . . . . 6 |- ( x = y -> B = C )
7	5, 6	eleq12d 2302	. . . . 5 |- ( x = y -> ( ( f ` x ) e. B <-> ( f ` y ) e. C ) )
8	2, 4, 7	cbvral 2763	. . . 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 2347	. 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 6868	. 2 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
12		dfixp 6868	. 2 |- X_ y e. A C = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
13	10, 11, 12	3eqtr4i 2262	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   F/_wnfc 2361   A.wral 2510   Fn wfn 5321   ` cfv 5326   X_cixp 6866

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fn 5329  df-fv 5334  df-ixp 6867

This theorem is referenced by:  cbvixpv  6884  mptelixpg  6902  prdsbas3  13369