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Metamath Proof Explorer

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 ) }