Theorem ixpeq1 6675

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 5277	. . . 4 |- ( A = B -> ( f Fn A <-> f Fn B ) )
2		raleq 2661	. . . 4 |- ( A = B -> ( A. x e. A ( f ` x ) e. C <-> A. x e. B ( f ` x ) e. C ) )
3	1, 2	anbi12d 465	. . 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 2284	. 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 6666	. 2 |- X_ x e. A C = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) }
6		dfixp 6666	. 2 |- X_ x e. B C = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) }
7	4, 5, 6	3eqtr4g 2224	1 |- ( A = B -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   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-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-fn 5191  df-ixp 6665

This theorem is referenced by:  ixpeq1d  6676