Theorem fconst 6091

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 5163	. . 3 |- ( A X. { B } ) = ( x e. A |-> B )
3	1, 2	fnmpti 6022	. 2 |- ( A X. { B } ) Fn A
4		rnxpss 5566	. 2 |- ran ( A X. { B } ) C_ { B }
5		df-f 5892	. 2 |- ( ( A X. { B } ) : A --> { B } <-> ( ( A X. { B } ) Fn A /\ ran ( A X. { B } ) C_ { B } ) )
6	3, 4, 5	mpbir2an 955	1 |- ( A X. { B } ) : A --> { B }

Syntax hints:   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fconstg  6092  fodomr  8111  ofsubeq0  11017  ser0f  12854  hashgval  13120  hashinf  13122  hashfxnn0  13124  hashfOLD  13126  prodf1f  14624  pwssplit1  19059  psrbag0  19494  xkofvcn  21487  ibl0  23553  dvcmul  23707  dvcmulf  23708  dvexp  23716  elqaalem3  24076  basellem7  24813  basellem9  24815  axlowdimlem8  25829  axlowdimlem9  25830  axlowdimlem10  25831  axlowdimlem11  25832  axlowdimlem12  25833  0oo  27644  occllem  28162  ho01i  28687  nlelchi  28920  hmopidmchi  29010  eulerpartlemt  30433  plymul02  30623  breprexpnat  30712  noetalem3  31865  fullfunfnv  32053  fullfunfv  32054  poimirlem16  33425  poimirlem19  33428  poimirlem23  33432  poimirlem24  33433  poimirlem25  33434  poimirlem28  33437  poimirlem29  33438  poimirlem30  33439  poimirlem31  33440  poimirlem32  33441  ftc1anclem5  33489  lfl0f  34356  diophrw  37322  pwssplit4  37659  ofsubid  38523  dvsconst  38529  dvsid  38530  binomcxplemnn0  38548  binomcxplemnotnn0  38555  aacllem  42547