Theorem ixpeq1 6921

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 5426	. . . 4 |- ( A = B -> ( f Fn A <-> f Fn B ) )
2		raleq 2731	. . . 4 |- ( A = B -> ( A. x e. A ( f ` x ) e. C <-> A. x e. B ( f ` x ) e. C ) )
3	1, 2	anbi12d 473	. . 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 2350	. 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 6912	. 2 |- X_ x e. A C = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) }
6		dfixp 6912	. 2 |- X_ x e. B C = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) }
7	4, 5, 6	3eqtr4g 2289	1 |- ( A = B -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   Fn wfn 5328   ` cfv 5333   X_cixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-fn 5336  df-ixp 6911

This theorem is referenced by:  ixpeq1d  6922