Theorem rnxrnres

Description: Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		rnxrn	⊢ ran ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) }
2		brres	⊢ ( 𝑦 ∈ V → ( 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝑦 ) ) )
3	2	elv	⊢ ( 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝑦 ) )
4	3	anbi2i	⊢ ( ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) ↔ ( 𝑢 𝑅 𝑥 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝑦 ) ) )
5		an12	⊢ ( ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) ) ↔ ( 𝑢 𝑅 𝑥 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝑦 ) ) )
6	4 5	bitr4i	⊢ ( ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) ↔ ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) ) )
7	6	exbii	⊢ ( ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) ) )
8		df-rex	⊢ ( ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) ) )
9	7 8	bitr4i	⊢ ( ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) )
10	9	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) }
11	1 10	eqtri	⊢ ran ( 𝑅 ⋉ ( 𝑆 ↾ 𝐴 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑆 𝑦 ) }