Theorem fconstg 5384

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 3587	. . . 4 |- ( x = B -> { x } = { B } )
2	1	xpeq2d 4628	. . 3 |- ( x = B -> ( A X. { x } ) = ( A X. { B } ) )
3		feq1 5320	. . . 4 |- ( ( A X. { x } ) = ( A X. { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { x } ) )
4		feq3 5322	. . . 4 |- ( { x } = { B } -> ( ( A X. { B } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
5	3, 4	sylan9bb 458	. . 3 |- ( ( ( A X. { x } ) = ( A X. { B } ) /\ { x } = { B } ) -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
6	2, 1, 5	syl2anc 409	. 2 |- ( x = B -> ( ( A X. { x } ) : A --> { x } <-> ( A X. { B } ) : A --> { B } ) )
7		vex 2729	. . 3 |- x e. _V
8	7	fconst 5383	. 2 |- ( A X. { x } ) : A --> { x }
9	6, 8	vtoclg 2786	1 |- ( B e. V -> ( A X. { B } ) : A --> { B } )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   {csn 3576   X. cxp 4602   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  fnconstg  5385  fconst6g  5386  xpsng  5660  fvconst2g  5699  fconst2g  5700  dvef  13328