Theorem ixpeq1 6763

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 5343	. . . 4 |- ( A = B -> ( f Fn A <-> f Fn B ) )
2		raleq 2690	. . . 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 2311	. 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 6754	. 2 |- X_ x e. A C = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. C ) }
6		dfixp 6754	. 2 |- X_ x e. B C = { f | ( f Fn B /\ A. x e. B ( f ` x ) e. C ) }
7	4, 5, 6	3eqtr4g 2251	1 |- ( A = B -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   Fn wfn 5249   ` cfv 5254   X_cixp 6752

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-fn 5257  df-ixp 6753

This theorem is referenced by:  ixpeq1d  6764