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  5843  fconstmpo  6116  ofc12  6259  caofinvl  6261  xpexgALT  6295  inftonninf  10705  fser0const  10798  prod1dc  12165  pws0g  13552  psrlinv  14717  psr1clfi  14721  mpl0fi  14735  cnmptc  15025  dvexp  15454  dvexp2  15455  dvmptidcn  15457  dvmptccn  15458  dvmptid  15459  dvmptc  15460  dvmptfsum  15468  dvef  15470  elply2  15478  plyconst  15488  plycolemc  15501  nninfall  16662  nninfsellemeqinf  16669  nninfnfiinf  16676  exmidsbthrlem  16677