Theorem ixpeq1 6603

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 5212	. . . 4 |- ( A = B -> ( f Fn A <-> f Fn B ) )
2		raleq 2626	. . . 4 |- ( A = B -> ( A. x e. A ( f ` x ) e. C <-> A. x e. B ( f ` x ) e. C ) )
3	1, 2	anbi12d 464	. . 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 2257	. 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 6594	. 2 |- X_ x e. A C = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) }
6		dfixp 6594	. 2 |- X_ x e. B C = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) }
7	4, 5, 6	3eqtr4g 2197	1 |- ( A = B -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   Fn wfn 5118   ` cfv 5123   X_cixp 6592

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-fn 5126  df-ixp 6593

This theorem is referenced by:  ixpeq1d  6604