Theorem fconst 5753

Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Hypothesis

Ref	Expression
fconst.1	|- B e. _V

Assertion

Ref	Expression
fconst	|- ( A X. { B } ) : A --> { B }

Proof of Theorem fconst

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconst.1	. . 3 |- B e. _V
2		fconstmpt 5032	. . 3 |- ( A X. { B } ) = ( x e. A |-> B )
3	1, 2	fnmpti 5691	. 2 |- ( A X. { B } ) Fn A
4		rnxpss 5424	. 2 |- ran ( A X. { B } ) C_ { B }
5		df-f 5574	. 2 |- ( ( A X. { B } ) : A --> { B } <-> ( ( A X. { B } ) Fn A /\ ran ( A X. { B } ) C_ { B } ) )
6	3, 4, 5	mpbir2an 918	1 |- ( A X. { B } ) : A --> { B }

Syntax hints:   e. wcel 1823   _Vcvv 3106   C_ wss 3461   {csn 4016   X. cxp 4986   ran crn 4989   Fn wfn 5565   -->wf 5566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574

This theorem is referenced by:  fconstg  5754  fodomr  7661  ofsubeq0  10528  ser0f  12142  hashgval  12390  hashinf  12392  hashf  12394  prodf1f  13783  pwssplit1  17900  psrbag0  18354  xkofvcn  20351  ibl0  22359  dvcmul  22513  dvcmulf  22514  dvexp  22522  elqaalem3  22883  basellem7  23558  basellem9  23560  axlowdimlem8  24454  axlowdimlem9  24455  axlowdimlem10  24456  axlowdimlem11  24457  axlowdimlem12  24458  0oo  25902  occllem  26419  ho01i  26945  nlelchi  27178  hmopidmchi  27268  eulerpartlemt  28574  plymul02  28767  fullfunfnv  29824  fullfunfv  29825  ftc1anclem5  30334  diophrw  30931  pwssplit4  31274  ofsubid  31470  dvsconst  31476  dvsid  31477  binomcxplemnn0  31495  binomcxplemnotnn0  31502  aacllem  33604  lfl0f  35191