Theorem fconstmpt 4802

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 3711	. . . 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 4182	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4760	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4178	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2265	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   {csn 3694   {copab 4175   |-> cmpt 4176   X. cxp 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sn 3700  df-opab 4177  df-mpt 4178  df-xp 4760

This theorem is referenced by:  fconst  5568  fcoconst  5853  fmptsn  5878  fconstmpo  6156  ofc12  6299  caofinvl  6301  xpexgALT  6339  inftonninf  10828  fser0const  10921  prod1dc  12297  pws0g  14155  rrgsupp  14512  psrlinv  14965  psr1clfi  14969  mpl0fi  14983  cnmptc  15273  dvexp  15702  dvexp2  15703  dvmptidcn  15705  dvmptccn  15706  dvmptid  15707  dvmptc  15708  dvmptfsum  15716  dvef  15718  elply2  15726  plyconst  15736  plycolemc  15749  nninfall  16913  nninfsellemeqinf  16920  nninfnfiinf  16927  exmidsbthrlem  16928