Theorem fconst 5694

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 4980	. . 3 |- ( A X. { B } ) = ( x e. A |-> B )
3	1, 2	fnmpti 5637	. 2 |- ( A X. { B } ) Fn A
4		rnxpss 5368	. 2 |- ran ( A X. { B } ) C_ { B }
5		df-f 5520	. 2 |- ( ( A X. { B } ) : A --> { B } <-> ( ( A X. { B } ) Fn A /\ ran ( A X. { B } ) C_ { B } ) )
6	3, 4, 5	mpbir2an 911	1 |- ( A X. { B } ) : A --> { B }

Syntax hints:   e. wcel 1758   _Vcvv 3068   C_ wss 3426   {csn 3975   X. cxp 4936   ran crn 4939   Fn wfn 5511   -->wf 5512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-fun 5518  df-fn 5519  df-f 5520

This theorem is referenced by:  fconstg  5695  fodomr  7562  ofsubeq0  10420  ser0f  11960  hashgval  12207  hashinf  12209  hashf  12211  pwssplit1  17246  psrbag0  17683  xkofvcn  19373  ibl0  21380  dvcmul  21534  dvcmulf  21535  dvexp  21543  elqaalem3  21903  basellem7  22540  basellem9  22542  axlowdimlem8  23330  axlowdimlem9  23331  axlowdimlem10  23332  axlowdimlem11  23333  axlowdimlem12  23334  0oo  24324  occllem  24841  ho01i  25367  nlelchi  25600  hmopidmchi  25690  eulerpartlemt  26888  plymul02  27081  prodf1f  27541  fullfunfnv  28111  fullfunfv  28112  ftc1anclem5  28609  diophrw  29235  pwssplit4  29580  ofsubid  29736  dvsconst  29742  dvsid  29743  lfl0f  33020