Theorem fconst 5523

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 4766	. . 3 |- ( A X. { B } ) = ( x e. A |-> B )
3	1, 2	fnmpti 5452	. 2 |- ( A X. { B } ) Fn A
4		rnxpss 5160	. 2 |- ran ( A X. { B } ) C_ { B }
5		df-f 5322	. 2 |- ( ( A X. { B } ) : A --> { B } <-> ( ( A X. { B } ) Fn A /\ ran ( A X. { B } ) C_ { B } ) )
6	3, 4, 5	mpbir2an 948	1 |- ( A X. { B } ) : A --> { B }

Syntax hints:   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666   X. cxp 4717   ran crn 4720   Fn wfn 5313   -->wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322

This theorem is referenced by:  fconstg  5524  exmidfodomrlemim  7387  ser0f  10764  prodf1f  12062  dvexp  15393