Theorem ndmovg

Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008)

	Ref	Expression
Assertion	ndmovg	⊢ ( ( dom 𝐹 = ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
2		eleq2	⊢ ( dom 𝐹 = ( 𝑅 × 𝑆 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 × 𝑆 ) ) )
3		opelxp	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 × 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) )
4	2 3	bitrdi	⊢ ( dom 𝐹 = ( 𝑅 × 𝑆 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
5	4	notbid	⊢ ( dom 𝐹 = ( 𝑅 × 𝑆 ) → ( ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ↔ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
6		ndmfv	⊢ ( ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∅ )
7	5 6	biimtrrdi	⊢ ( dom 𝐹 = ( 𝑅 × 𝑆 ) → ( ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∅ ) )
8	7	imp	⊢ ( ( dom 𝐹 = ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∅ )
9	1 8	eqtrid	⊢ ( ( dom 𝐹 = ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ∅ )