Theorem nssdmovg

Description: The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017)

	Ref	Expression
Assertion	nssdmovg	⊢ ( ( 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		ssel2	⊢ ( ( 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	sylib	⊢ ( ( dom 𝐹 ⊆ ( 𝑅 × 𝑆 ) ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ) → ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) )
5	4	stoic1a	⊢ ( ( dom 𝐹 ⊆ ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 )
6		ndmfv	⊢ ( ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∅ )
7	5 6	syl	⊢ ( ( dom 𝐹 ⊆ ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∅ )
8	1 7	eqtrid	⊢ ( ( dom 𝐹 ⊆ ( 𝑅 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ∅ )