Theorem fmptsn

Description: Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006) (Proof shortened by Andrew Salmon, 22-Oct-2011) (Revised by Stefan O'Rear, 28-Feb-2015)

	Ref	Expression
Assertion	fmptsn	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{⟨A,B⟩\right\}=\left(x\in \left\{A\right\}⟼B\right)$$

Proof

Step	Hyp	Ref	Expression
1		xpsng	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A\right\}\times \left\{B\right\}=\left\{⟨A,B⟩\right\}$$
2		fconstmpt	$$⊢\left\{A\right\}\times \left\{B\right\}=\left(x\in \left\{A\right\}⟼B\right)$$
3	1 2	eqtr3di	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{⟨A,B⟩\right\}=\left(x\in \left\{A\right\}⟼B\right)$$