Theorem cbvixp 6825

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 2363	. . . . 5 |- F/ y ( f ` x ) e. B
3		cbvixp.2	. . . . . 6 |- F/_ x C
4	3	nfel2 2363	. . . . 5 |- F/ x ( f ` y ) e. C
5		fveq2 5599	. . . . . 6 |- ( x = y -> ( f ` x ) = ( f ` y ) )
6		cbvixp.3	. . . . . 6 |- ( x = y -> B = C )
7	5, 6	eleq12d 2278	. . . . 5 |- ( x = y -> ( ( f ` x ) e. B <-> ( f ` y ) e. C ) )
8	2, 4, 7	cbvral 2738	. . . 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 2323	. 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 6810	. 2 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
12		dfixp 6810	. 2 |- X_ y e. A C = { f | ( f Fn A /\ A. y e. A ( f ` y ) e. C ) }
13	10, 11, 12	3eqtr4i 2238	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   F/_wnfc 2337   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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  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-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fn 5293  df-fv 5298  df-ixp 6809

This theorem is referenced by:  cbvixpv  6826  mptelixpg  6844  prdsbas3  13234