Theorem elres

Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011) (Proof shortened by Peter Mazsa, 9-Sep-2022)

	Ref	Expression
Assertion	elres	⊢ ( 𝐴 ∈ ( 𝐵 ↾ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-res	⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) )
2	1	eleq2i	⊢ ( 𝐴 ∈ ( 𝐵 ↾ 𝐶 ) ↔ 𝐴 ∈ ( 𝐵 ∩ ( 𝐶 × V ) ) )
3		elinxp	⊢ ( 𝐴 ∈ ( 𝐵 ∩ ( 𝐶 × V ) ) ↔ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ V ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
4		rexv	⊢ ( ∃ 𝑦 ∈ V ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ↔ ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
5	4	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ V ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
6	2 3 5	3bitri	⊢ ( 𝐴 ∈ ( 𝐵 ↾ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )