Theorem fconstmpt 5682

Description: Representation of a constant function using the mapping operation. (Note that x cannot appear free in B.) (Contributed by set.mm contributors, 16-Nov-2013.)

Assertion

Ref	Expression
fconstmpt	|- (A X. {B}) = (x e. A |-> B)

Distinct variable groups:   x,A   x,B

Proof of Theorem fconstmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstopab 4816	. 2 |- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}
2		df-mpt 5653	. 2 |- (x e. A |-> B) = {<.x, y>. | (x e. A /\ y = B)}
3	1, 2	eqtr4i 2376	1 |- (A X. {B}) = (x e. A |-> B)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  {csn 3738  {copab 4623   X. cxp 4771   |-> cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3742  df-opab 4624  df-xp 4785  df-mpt 5653

This theorem is referenced by: (None)