Theorem relssres

Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	relssres	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↾ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → Rel 𝐴 )
2		vex	⊢ 𝑥 ∈ V
3		vex	⊢ 𝑦 ∈ V
4	2 3	opeldm	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴 )
5		ssel	⊢ ( dom 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ dom 𝐴 → 𝑥 ∈ 𝐵 ) )
6	4 5	syl5	⊢ ( dom 𝐴 ⊆ 𝐵 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
7	6	ancrd	⊢ ( dom 𝐴 ⊆ 𝐵 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) ) )
8	3	opelresi	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) )
9	7 8	imbitrrdi	⊢ ( dom 𝐴 ⊆ 𝐵 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ) )
10	9	adantl	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ) )
11	1 10	relssdv	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ ( 𝐴 ↾ 𝐵 ) )
12		resss	⊢ ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴
13	11 12	jctil	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → ( ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴 ∧ 𝐴 ⊆ ( 𝐴 ↾ 𝐵 ) ) )
14		eqss	⊢ ( ( 𝐴 ↾ 𝐵 ) = 𝐴 ↔ ( ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴 ∧ 𝐴 ⊆ ( 𝐴 ↾ 𝐵 ) ) )
15	13 14	sylibr	⊢ ( ( Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↾ 𝐵 ) = 𝐴 )