Theorem fconst

Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Hypothesis	fconst.1	$$⊢B\in \mathrm{V}$$
Assertion	fconst	$$⊢\left(A\times \left\{B\right\}\right):A⟶\left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		fconst.1	$$⊢B\in \mathrm{V}$$
2		fconstmpt	$$⊢A\times \left\{B\right\}=\left(x\in A⟼B\right)$$
3	1 2	fnmpti	$$⊢\left(A\times \left\{B\right\}\right)FnA$$
4		rnxpss	$$⊢\mathrm{ran}\left(A\times \left\{B\right\}\right)\subseteq \left\{B\right\}$$
5		df-f	$$⊢\left(A\times \left\{B\right\}\right):A⟶\left\{B\right\}\leftrightarrow \left(\left(A\times \left\{B\right\}\right)FnA\wedge \mathrm{ran}\left(A\times \left\{B\right\}\right)\subseteq \left\{B\right\}\right)$$
6	3 4 5	mpbir2an	$$⊢\left(A\times \left\{B\right\}\right):A⟶\left\{B\right\}$$