Theorem xpima2

Description: Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017)

	Ref	Expression
Assertion	xpima2	⊢ ( ( 𝐴 ∩ 𝐶 ) ≠ ∅ → ( ( 𝐴 × 𝐵 ) “ 𝐶 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		xpima	⊢ ( ( 𝐴 × 𝐵 ) “ 𝐶 ) = if ( ( 𝐴 ∩ 𝐶 ) = ∅ , ∅ , 𝐵 )
2		ifnefalse	⊢ ( ( 𝐴 ∩ 𝐶 ) ≠ ∅ → if ( ( 𝐴 ∩ 𝐶 ) = ∅ , ∅ , 𝐵 ) = 𝐵 )
3	1 2	eqtrid	⊢ ( ( 𝐴 ∩ 𝐶 ) ≠ ∅ → ( ( 𝐴 × 𝐵 ) “ 𝐶 ) = 𝐵 )