This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fconstg

Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004)

Ref Expression
Assertion fconstg ( 𝐵 ∈ 𝑉 → ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝐵 } )

Proof

Step Hyp Ref Expression
1 sneq ⊢ ( 𝑥 = 𝐵 → { 𝑥 } = { 𝐵 } )
2 1 xpeq2d ⊢ ( 𝑥 = 𝐵 → ( 𝐴 × { 𝑥 } ) = ( 𝐴 × { 𝐵 } ) )
3 feq1 ⊢ ( ( 𝐴 × { 𝑥 } ) = ( 𝐴 × { 𝐵 } ) → ( ( 𝐴 × { 𝑥 } ) : 𝐴 ⟶ { 𝑥 } ↔ ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝑥 } ) )
4 feq3 ⊢ ( { 𝑥 } = { 𝐵 } → ( ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝑥 } ↔ ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝐵 } ) )
5 3 4 sylan9bb ⊢ ( ( ( 𝐴 × { 𝑥 } ) = ( 𝐴 × { 𝐵 } ) ∧ { 𝑥 } = { 𝐵 } ) → ( ( 𝐴 × { 𝑥 } ) : 𝐴 ⟶ { 𝑥 } ↔ ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝐵 } ) )
6 2 1 5 syl2anc ⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 × { 𝑥 } ) : 𝐴 ⟶ { 𝑥 } ↔ ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝐵 } ) )
7 vex ⊢ 𝑥 ∈ V
8 7 fconst ⊢ ( 𝐴 × { 𝑥 } ) : 𝐴 ⟶ { 𝑥 }
9 6 8 vtoclg ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 × { 𝐵 } ) : 𝐴 ⟶ { 𝐵 } )