Theorem bj-rest0

Description: An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021)

	Ref	Expression
Assertion	bj-rest0	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( ∅ ∈ 𝑋 → ∅ ∈ ( 𝑋 ↾t 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		in0	⊢ ( 𝐴 ∩ ∅ ) = ∅
2		incom	⊢ ( 𝐴 ∩ ∅ ) = ( ∅ ∩ 𝐴 )
3	1 2	eqtr3i	⊢ ∅ = ( ∅ ∩ 𝐴 )
4		0ex	⊢ ∅ ∈ V
5		eleq1	⊢ ( 𝑥 = ∅ → ( 𝑥 ∈ 𝑋 ↔ ∅ ∈ 𝑋 ) )
6		ineq1	⊢ ( 𝑥 = ∅ → ( 𝑥 ∩ 𝐴 ) = ( ∅ ∩ 𝐴 ) )
7	6	eqeq2d	⊢ ( 𝑥 = ∅ → ( ∅ = ( 𝑥 ∩ 𝐴 ) ↔ ∅ = ( ∅ ∩ 𝐴 ) ) )
8	5 7	anbi12d	⊢ ( 𝑥 = ∅ → ( ( 𝑥 ∈ 𝑋 ∧ ∅ = ( 𝑥 ∩ 𝐴 ) ) ↔ ( ∅ ∈ 𝑋 ∧ ∅ = ( ∅ ∩ 𝐴 ) ) ) )
9	4 8	spcev	⊢ ( ( ∅ ∈ 𝑋 ∧ ∅ = ( ∅ ∩ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ ∅ = ( 𝑥 ∩ 𝐴 ) ) )
10	3 9	mpan2	⊢ ( ∅ ∈ 𝑋 → ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ ∅ = ( 𝑥 ∩ 𝐴 ) ) )
11		df-rex	⊢ ( ∃ 𝑥 ∈ 𝑋 ∅ = ( 𝑥 ∩ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ ∅ = ( 𝑥 ∩ 𝐴 ) ) )
12	10 11	sylibr	⊢ ( ∅ ∈ 𝑋 → ∃ 𝑥 ∈ 𝑋 ∅ = ( 𝑥 ∩ 𝐴 ) )
13		elrest	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( ∅ ∈ ( 𝑋 ↾t 𝐴 ) ↔ ∃ 𝑥 ∈ 𝑋 ∅ = ( 𝑥 ∩ 𝐴 ) ) )
14	12 13	syl5ibr	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( ∅ ∈ 𝑋 → ∅ ∈ ( 𝑋 ↾t 𝐴 ) ) )