Theorem fconstg 5450

Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
fconstg	|- ( B e. V -> ( A X. { B } ) : A --> { B } )

Proof of Theorem fconstg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3629	. . . 4 |- ( x = B -> { x } = { B } )
2	1	xpeq2d 4683	. . 3 |- ( x = B -> ( A X. { x } ) = ( A X. { B } ) )
3		feq1 5386	. . . 4 |- ( ( A X. { x } ) = ( A X. { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { x } ) )
4		feq3 5388	. . . 4 |- ( { x } = { B } -> ( ( A X. { B } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
5	3, 4	sylan9bb 462	. . 3 |- ( ( ( A X. { x } ) = ( A X. { B } ) /\ { x } = { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
6	2, 1, 5	syl2anc 411	. 2 |- ( x = B -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
7		vex 2763	. . 3 |- x e. _V
8	7	fconst 5449	. 2 |- ( A X. { x } ) : A --> { x }
9	6, 8	vtoclg 2820	1 |- ( B e. V -> ( A X. { B } ) : A --> { B } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {csn 3618   X. cxp 4657   -->wf 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  fnconstg  5451  fconst6g  5452  xpsng  5733  fvconst2g  5772  fconst2g  5773  dvef  14873