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Metamath Proof Explorer

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 )