Theorem fconstmpt 4817

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 3722	. . . 4 |- ( y e. { B } <-> y = B )
2	1	anbi2i 461	. . 3 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
3	2	opabbii 4193	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4775	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4189	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2269	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3705   {copab 4186   |-> cmpt 4187   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711  df-opab 4188  df-mpt 4189  df-xp 4775

This theorem is referenced by:  fconst  5583  fcoconst  5870  fmptsn  5895  fconstmpo  6173  ofc12  6316  caofinvl  6318  xpexgALT  6356  inftonninf  10857  fser0const  10950  prod1dc  12331  gsumconstcmn  14143  pws0g  14190  rrgsupp  14547  psrlinv  14998  psr1clfi  15002  mpl0fi  15016  cnmptc  15306  dvexp  15735  dvexp2  15736  dvmptidcn  15738  dvmptccn  15739  dvmptid  15740  dvmptc  15741  dvmptfsum  15749  dvef  15751  elply2  15759  plyconst  15769  plycolemc  15782  nninfall  16957  nninfsellemeqinf  16964  nninfnfiinf  16971  exmidsbthrlem  16972