Theorem ixpeq1 6533

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq1	|- ( A = B -> X_ x e. A C = X_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem ixpeq1

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq2 5148	. . . 4 |- ( A = B -> ( f Fn A <-> f Fn B ) )
2		raleq 2584	. . . 4 |- ( A = B -> ( A. x e. A ( f ` x ) e. C <-> A. x e. B ( f ` x ) e. C ) )
3	1, 2	anbi12d 460	. . 3 |- ( A = B -> ( ( f Fn A /\ A. x e. A ( f ` x ) e. C ) <-> ( f Fn B /\ A. x e. B ( f ` x ) e. C ) ) )
4	3	abbidv 2217	. 2 |- ( A = B -> { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) } = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) } )
5		dfixp 6524	. 2 |- X_ x e. A C = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) }
6		dfixp 6524	. 2 |- X_ x e. B C = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) }
7	4, 5, 6	3eqtr4g 2157	1 |- ( A = B -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   {cab 2086   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-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-fn 5062  df-ixp 6523

This theorem is referenced by:  ixpeq1d  6534