Theorem fconstmpt 4797

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 3706	. . . 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 4177	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4755	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4173	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2263	1 |- ( A X. { B } ) = ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2203   {csn 3689   {copab 4170   |-> cmpt 4171   X. cxp 4747

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sn 3695  df-opab 4172  df-mpt 4173  df-xp 4755

This theorem is referenced by:  fconst  5563  fcoconst  5848  fmptsn  5873  fconstmpo  6148  ofc12  6290  caofinvl  6292  xpexgALT  6326  inftonninf  10804  fser0const  10897  prod1dc  12272  pws0g  13664  rrgsupp  14411  psrlinv  14839  psr1clfi  14843  mpl0fi  14857  cnmptc  15147  dvexp  15576  dvexp2  15577  dvmptidcn  15579  dvmptccn  15580  dvmptid  15581  dvmptc  15582  dvmptfsum  15590  dvef  15592  elply2  15600  plyconst  15610  plycolemc  15623  nninfall  16787  nninfsellemeqinf  16794  nninfnfiinf  16801  exmidsbthrlem  16802