Theorem ixpeq1 4353

Description: Equality theorem for infinite Cartesian product.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ixpeq1

Step	Hyp	Ref	Expression
1		fneq2 3583	. . . 4 |- (A = B -> (f Fn A <-> f Fn B))
2		raleq1 1786	. . . 4 |- (A = B -> (A.x e. A (f` x) e. C <-> A.x e. B (f` x) e. C))
3	1, 2	anbi12d 628	. . 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 1577	. 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		df-ixp 4348	. 2 |- X_x e. A C = {f | (f Fn A /\ A.x e. A (f` x) e. C)}
6		df-ixp 4348	. 2 |- X_x e. B C = {f | (f Fn B /\ A.x e. B (f` x) e. C)}
7	4, 5, 6	3eqtr4g 1531	1 |- (A = B -> X_x e. A C = X_x e. B C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645   Fn wfn 3177  ` cfv 3182  X_cixp 4347

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-fn 3193  df-ixp 4348

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