Theorem fconstmpt 4633

Description: Representation of a constant function using the mapping operation. (Note that x cannot appear free in B.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 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		velsn 3577	. . . 4 |- ( y e. { B } <-> y = B )
2	1	anbi2i 453	. . 3 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
3	2	opabbii 4031	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4592	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4027	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2188	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1335   e. wcel 2128   {csn 3560   {copab 4024   |-> cmpt 4025   X. cxp 4584

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sn 3566  df-opab 4026  df-mpt 4027  df-xp 4592

This theorem is referenced by:  fconst  5365  fcoconst  5638  fmptsn  5656  fconstmpo  5916  ofc12  6052  caofinvl  6054  xpexgALT  6081  inftonninf  10340  fser0const  10415  prod1dc  11483  cnmptc  12682  dvexp  13075  dvexp2  13076  dvmptidcn  13078  dvmptccn  13079  dvef  13088  nninfall  13581  nninfsellemeqinf  13588  exmidsbthrlem  13593