Theorem rnxrnidres

Description: Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021)

	Ref	Expression
Assertion	rnxrnidres	⊢ ran ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 = 𝑦 ∧ 𝑢 𝑅 𝑥 ) }

Proof

Step	Hyp	Ref	Expression
1		rnxrnres	⊢ ran ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 I 𝑦 ) }
2		ideqg	⊢ ( 𝑦 ∈ V → ( 𝑢 I 𝑦 ↔ 𝑢 = 𝑦 ) )
3	2	elv	⊢ ( 𝑢 I 𝑦 ↔ 𝑢 = 𝑦 )
4	3	anbi1ci	⊢ ( ( 𝑢 𝑅 𝑥 ∧ 𝑢 I 𝑦 ) ↔ ( 𝑢 = 𝑦 ∧ 𝑢 𝑅 𝑥 ) )
5	4	rexbii	⊢ ( ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 I 𝑦 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 = 𝑦 ∧ 𝑢 𝑅 𝑥 ) )
6	5	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 I 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 = 𝑦 ∧ 𝑢 𝑅 𝑥 ) }
7	1 6	eqtri	⊢ ran ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 = 𝑦 ∧ 𝑢 𝑅 𝑥 ) }