Theorem fconstg 5220

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 3461	. . . 4 |- ( x = B -> { x } = { B } )
2	1	xpeq2d 4475	. . 3 |- ( x = B -> ( A X. { x } ) = ( A X. { B } ) )
3		feq1 5158	. . . 4 |- ( ( A X. { x } ) = ( A X. { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { x } ) )
4		feq3 5160	. . . 4 |- ( { x } = { B } -> ( ( A X. { B } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
5	3, 4	sylan9bb 451	. . 3 |- ( ( ( A X. { x } ) = ( A X. { B } ) /\ { x } = { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
6	2, 1, 5	syl2anc 404	. 2 |- ( x = B -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
7		vex 2623	. . 3 |- x e. _V
8	7	fconst 5219	. 2 |- ( A X. { x } ) : A --> { x }
9	6, 8	vtoclg 2680	1 |- ( B e. V -> ( A X. { B } ) : A --> { B } )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   e. wcel 1439   {csn 3450   X. cxp 4449   -->wf 5024

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-f 5032

This theorem is referenced by:  fnconstg  5221  fconst6g  5222  xpsng  5486  fvconst2g  5525  fconst2g  5526