Theorem fconstmpt 4824

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 3726	. . . 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 4199	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4782	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4195	. 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 )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3709   {copab 4192   |-> cmpt 4193   X. cxp 4774

This proof depends on 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 proof 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 3715  df-opab 4194  df-mpt 4195  df-xp 4782

This theorem is used by:  fconst  5591  fcoconst  5882  fmptsn  5907  fconstmpo  6186  ofc12  6330  caofinvl  6332  xpexgALT  6370  inftonninf  10900  fser0const  10993  prod1dc  12378  gsumconstcmn  14266  pws0g  14320  rrgsupp  14683  psrlinv  15194  psr1clfi  15198  psrlidmfi  15199  psrridmfi  15200  mpl0fi  15218  cnmptc  15508  dvexp  15937  dvexp2  15938  dvmptidcn  15940  dvmptccn  15941  dvmptid  15942  dvmptc  15943  dvmptfsum  15951  dvef  15953  elply2  15961  plyconst  15971  plycolemc  15984  nninfall  17258  nninfsellemeqinf  17265  nninfnfiinf  17272  exmidsbthrlem  17273