Theorem br1cossxrncnvssrres

Description: <. B , C >. and <. D , E >. are cosets by range Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021)

	Ref	Expression
Assertion	br1cossxrncnvssrres	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) ∧ ( 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) ) → ( ⟨ 𝐵 , 𝐶 ⟩ ≀ ( 𝑅 ⋉ ( ◡ S ↾ 𝐴 ) ) ⟨ 𝐷 , 𝐸 ⟩ ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐶 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐸 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐷 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		br1cossxrnres	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) ∧ ( 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) ) → ( ⟨ 𝐵 , 𝐶 ⟩ ≀ ( 𝑅 ⋉ ( ◡ S ↾ 𝐴 ) ) ⟨ 𝐷 , 𝐸 ⟩ ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝑢 ◡ S 𝐶 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ S 𝐸 ∧ 𝑢 𝑅 𝐷 ) ) ) )
2		brcnvssr	⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢 ) )
3	2	elv	⊢ ( 𝑢 ◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢 )
4	3	anbi1i	⊢ ( ( 𝑢 ◡ S 𝐶 ∧ 𝑢 𝑅 𝐵 ) ↔ ( 𝐶 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐵 ) )
5		brcnvssr	⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢 ) )
6	5	elv	⊢ ( 𝑢 ◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢 )
7	6	anbi1i	⊢ ( ( 𝑢 ◡ S 𝐸 ∧ 𝑢 𝑅 𝐷 ) ↔ ( 𝐸 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐷 ) )
8	4 7	anbi12i	⊢ ( ( ( 𝑢 ◡ S 𝐶 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ S 𝐸 ∧ 𝑢 𝑅 𝐷 ) ) ↔ ( ( 𝐶 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐸 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐷 ) ) )
9	8	rexbii	⊢ ( ∃ 𝑢 ∈ 𝐴 ( ( 𝑢 ◡ S 𝐶 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ◡ S 𝐸 ∧ 𝑢 𝑅 𝐷 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐶 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐸 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐷 ) ) )
10	1 9	bitrdi	⊢ ( ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) ∧ ( 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) ) → ( ⟨ 𝐵 , 𝐶 ⟩ ≀ ( 𝑅 ⋉ ( ◡ S ↾ 𝐴 ) ) ⟨ 𝐷 , 𝐸 ⟩ ↔ ∃ 𝑢 ∈ 𝐴 ( ( 𝐶 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝐸 ⊆ 𝑢 ∧ 𝑢 𝑅 𝐷 ) ) ) )