Theorem fconstmpt 4581

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 3539	. . . 4 |- ( y e. { B } <-> y = B )
2	1	anbi2i 452	. . 3 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
3	2	opabbii 3990	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4540	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 3986	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2168	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   {csn 3522   {copab 3983   |-> cmpt 3984   X. cxp 4532

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528  df-opab 3985  df-mpt 3986  df-xp 4540

This theorem is referenced by:  fconst  5313  fcoconst  5584  fmptsn  5602  fconstmpo  5859  ofc12  5995  caofinvl  5997  xpexgALT  6024  inftonninf  10207  fser0const  10282  cnmptc  12440  dvexp  12833  dvexp2  12834  dvmptidcn  12836  dvmptccn  12837  dvef  12845  nninfall  13193  nninfsellemeqinf  13201  nninffeq  13205  exmidsbthrlem  13206