Theorem fconst 5541

Description: A cross 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 4779	. . 3 |- ( A X. { B } ) = ( x e. A |-> B )
3	1, 2	fnmpti 5468	. 2 |- ( A X. { B } ) Fn A
4		rnxpss 5175	. 2 |- ran ( A X. { B } ) C_ { B }
5		df-f 5337	. 2 |- ( ( A X. { B } ) : A --> { B } <-> ( ( A X. { B } ) Fn A /\ ran ( A X. { B } ) C_ { B } ) )
6	3, 4, 5	mpbir2an 951	1 |- ( A X. { B } ) : A --> { B }

Syntax hints:   e. wcel 2202   _Vcvv 2803   C_ wss 3201   {csn 3673   X. cxp 4729   ran crn 4732   Fn wfn 5328   -->wf 5329

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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  fconstg  5542  exmidfodomrlemim  7455  ser0f  10842  prodf1f  12167  dvexp  15505