Theorem fconstg 5454

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 3633	. . . 4 |- ( x = B -> { x } = { B } )
2	1	xpeq2d 4687	. . 3 |- ( x = B -> ( A X. { x } ) = ( A X. { B } ) )
3		feq1 5390	. . . 4 |- ( ( A X. { x } ) = ( A X. { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { x } ) )
4		feq3 5392	. . . 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 2766	. . 3 |- x e. _V
8	7	fconst 5453	. 2 |- ( A X. { x } ) : A --> { x }
9	6, 8	vtoclg 2824	1 |- ( B e. V -> ( A X. { B } ) : A --> { B } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {csn 3622   X. cxp 4661   -->wf 5254

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262

This theorem is referenced by:  fnconstg  5455  fconst6g  5456  xpsng  5737  fvconst2g  5776  fconst2g  5777  dvef  14963