Theorem xpimasn

Description: Direct image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018) (Proof shortened by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	xpimasn	⊢ ( 𝑋 ∈ 𝐴 → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		disjsn	⊢ ( ( 𝐴 ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ 𝐴 )
2	1	necon3abii	⊢ ( ( 𝐴 ∩ { 𝑋 } ) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴 )
3		notnotb	⊢ ( 𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴 )
4	2 3	bitr4i	⊢ ( ( 𝐴 ∩ { 𝑋 } ) ≠ ∅ ↔ 𝑋 ∈ 𝐴 )
5		xpima2	⊢ ( ( 𝐴 ∩ { 𝑋 } ) ≠ ∅ → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = 𝐵 )
6	4 5	sylbir	⊢ ( 𝑋 ∈ 𝐴 → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = 𝐵 )