Theorem fconstmpt 4773

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 3686	. . . 4 |- ( y e. { B } <-> y = B )
2	1	anbi2i 457	. . 3 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
3	2	opabbii 4156	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4731	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4152	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2262	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   {csn 3669   {copab 4149   |-> cmpt 4150   X. cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sn 3675  df-opab 4151  df-mpt 4152  df-xp 4731

This theorem is referenced by:  fconst  5532  fcoconst  5818  fmptsn  5842  fconstmpo  6115  ofc12  6258  caofinvl  6260  xpexgALT  6294  inftonninf  10703  fser0const  10796  prod1dc  12146  pws0g  13533  psrlinv  14697  psr1clfi  14701  mpl0fi  14715  cnmptc  15005  dvexp  15434  dvexp2  15435  dvmptidcn  15437  dvmptccn  15438  dvmptid  15439  dvmptc  15440  dvmptfsum  15448  dvef  15450  elply2  15458  plyconst  15468  plycolemc  15481  nninfall  16611  nninfsellemeqinf  16618  nninfnfiinf  16625  exmidsbthrlem  16626